Mhand Hifi

Heuristic algorithms for the multiple-choice multidimensional knapsack problem

In this paper, we propose several heuristics for approximately solving the multiple-choice multidimensional knapsack problem (noted MMKP), an NP-Hard combinatorial optimization problem. The first algorithm is a constructive approach used especially for constructing an initial feasible solution for the problem. The second approach is applied in order to improve the quality of the initial solution. Finally, we introduce the main algorithm, which starts by applying the first approach and tries to produce a better solution to the MMKP. The last approach can be viewed as a two-stage procedure: (i) the first stage is applied in order to penalize a chosen feasible solution and, (ii) the second stage is used in order to normalize and to improve the solution given by the firs stage. The performance of the proposed approaches has been evaluated based problem instances extracted from the literature. Encouraging results have been obtained.

An improvement of Viswanathan and Bagchi's exact algorithm for constrained two-dimensional cutting stock

Abstract Viswanathan and Bagchi [ Operations Research , 1993, 41 (4), 768–776] [1] have proposed a bottom-up algorithm which combines in the nice tree-search procedure Gilmore and Gomorys algorithm, called at each node of the tree, for solving exactly the constrained two-dimensional cutting problem. This algorithm is one of the best exact algorithms known today. In this paper, we propose an improved version of this algorithm by introducing one-dimensional bounded knapsacks in the original algorithm. Then, by exploiting dynamic programming properties, we obtain good lower and upper bounds which lead to significant branching cuts. Finally, the improved version is compared to the standard version of Viswanathan and Bagchi on some small and medium instances.

Exact Algorithms for Large-Scale Unconstrained Two and Three Staged Cutting Problems

In this paper we propose two exact algorithms for solving both two-staged and three staged unconstrained (un)weighted cutting problems. The two-staged problem is solved by applying a dynamic programming procedure originally developed by Gilmore and Gomory [Gilmore and Gomory, Operations Research, vol. 13, pp. 94–119, 1965]. The three-staged problem is solved by using a top-down approach combined with a dynamic programming procedure. The performance of the exact algorithms are evaluated on some problem instances of the literature and other hard randomly-generated problem instances (a total of 53 problem instances). A parallel implementation is an important feature of the algorithm used for solving the three-staged version.

Approximate algorithms for constrained circular cutting problems

In this paper, we study the problem of cutting a rectangular plate R of dimensions (L, W) into as many circular pieces as possible. The circular pieces are of n different types with radii ri, i = 1 ,..., n. We solve the constrained circular problem, where di the maximum demand for piece type i is specified, using two heuristics: a constructive procedure-based heuristic and a genetic algorithm-based heuristic. Both of these approaches search for a good ordering of the pieces and use an adaptation of the best local position procedure (Studia. Inform. Univ. 2 (1) (2002) 33) to find the best layout of this ordered set. This positioning procedure is specifically tailored to circular cutting problems. It acts, for constrained problems, as one of the mutation operators of the genetic algorithm. We compare the performance of both proposed approaches to that of existing approximate and exact algorithms on several problem instances taken from the literature. The computational results show that the proposed approaches produce high-quality solutions within reasonable computational times. The genetic algorithm-based heuristic is easily parallelizable; one of its important features to be investigated in the near future.

Exact algorithms for the guillotine strip cutting/packing problem

Abstract The strip cutting/packing problem consists of cutting a large strip with a fixed-width and unlimited length into smaller subrectangles, without violating the demand values imposed on each subrectangle. Computer science, industrial engineering, logistics, manufacturing, management, production processes are among obvious fields of applications. In this paper we propose exact approaches for solving optimally the strip cutting/packing problem. The proposed algorithms are based upon branch-and-bound and dynamic programming procedures. In this paper we present exact algorithms for strip cutting/packing problems. The proposed algorithms are based upon branch-and-bound procedures. In this paper we present exact algorithms for strip cutting/packing problems. The proposed algorithms are based upon branch-and-bound procedures. In the first algorithm, the problem is reduced to a series of two-dimensional constrained cutting stock problems. Each step of this algorithm is solved by using a recent algorithm (called MVB) developed by Hifi [Hifi, M., An improvement of Viswanathan and Bagchis exact algorithm for cutting stock problems. Computers and Operations Research, 1997, 24(8), 727–736.]. The second algorithm considers a large stock rectangle, constructed by using a heuristic algorithm for limiting the length of the initial strip. Then, we apply the MVB algorithm by using the best-first search strategy. Computational results show that the proposed exact algorithms are able to solve some small and medium problem instances within reasonable execution times. These algorithms are easily parallelizable and this is one of their important futures.

A recursive exact algorithm for weighted two-dimensional cutting

Gilmore and Gomorys algorithm is one of the better actually known exact algorithms for solving unconstrained guillotine two-dimensional cutting problems. Herzs algorithm is more effective, but only for the unweighted case. We propose a new exact algorithm adequate for both weighted and unweighted cases, which is more powerful than both algorithms. The algorithm uses dynamic programming procedures and one-dimensional knapsack problem to obtain efficient lower and upper bounds and important optimality criteria which permit a significant branching cut in a recursive tree-search procedure. Recursivity, computational power, adequateness to parallel implementations, and generalization for solving constrained two-dimensional cutting problems, are some important features of the new algorithm.

Approximate and Exact Algorithms for Constrained (Un) Weighted Two-dimensional Two-staged Cutting Stock Problems

In this paper we propose two algorithms for solving both unweighted and weighted constrained two-dimensional two-staged cutting stock problems. The problem is called two-staged cutting problem because each produced (sub)optimal cutting pattern is realized by using two cut-phases. In the first cut-phase, the current stock rectangle is slit down its width (resp. length) into a set of vertical (resp. horizontal) strips and, in the second cut-phase, each of these strips is taken individually and chopped across its length (resp. width).First, we develop an approximate algorithm for the problem. The original problem is reduced to a series of single bounded knapsack problems and solved by applying a dynamic programming procedure. Second, we propose an exact algorithm tailored especially for the constrained two-staged cutting problem. The algorithm starts with an initial (feasible) lower bound computed by applying the proposed approximate algorithm. Then, by exploiting dynamic programming properties, we obtain good lower and upper bounds which lead to significant branching cuts. Extensive computational testing on problem instances from the literature shows the effectiveness of the proposed approximate and exact approaches.

A simulated annealing approach for the circular cutting problem

We propose a heuristic for the constrained and the unconstrained circular cutting problem based upon simulated annealing. We define an energy function, the small values of which provide a good concentration of the circular pieces on the left bottom corner of the initial rectangle. Such values of the energy correspond to configurations where pieces are placed in the rectangle without overlapping. Appropriate software has been devised and computational results and comparisons with some other algorithms are also provided and discussed.

The DH/KD algorithm: a hybrid approach for unconstrained two-dimensional cutting problems

Abstract We study both weighted and unweighted unconstrained two-dimensional guillotine cutting problems. We develop a hybrid approach which combines two heuristics from the literature. The first one (DH) uses a tree-search procedure introducing two strategies: Depth-first search and Hill-climbing. The second one (KD) is based on a series of one-dimensional Knapsack problems using Dynamic programming techniques. The DH /KD algorithm starts with a good initial lower bound obtained by using the KD algorithm. At each level of the tree-search, the proposed algorithm uses also the KD algorithm for constructing new lower bounds and uses another one-dimensional knapsack for constructing refinement upper bounds. The resulting algorithm can be seen as a generalization of the two heuristics and solves large problem instances very well within small computational time. Our algorithm is compared to Morabito et al.s algorithm (the unweighted case), and to Beasleys [2] approach (the weighted case) on some examples taken from the literature as well as randomly generated instances.

An Exact Algorithm for Constrained Two-Dimensional Two-Staged Cutting Problems

Theconstrained two-dimensional cutting (C_TDC) problem consists of determining acutting pattern of a set ofn small rectangular piece types on a rectangular stock plateS with lengthL and widthW, to maximize the sum of the profits of the pieces to be cut. Each piece typei,i=1,..., n, is characterized by a lengthl i , a widthw i , a profit (or weight)c i , and an upper demand valueb i . The upper demand value is the maximum number of pieces of typei that can be cut onS. In this paper, we study the two-staged C_TDC problem, noted C_2TDC. It is a classical variant of the C_TDC where each piece is produced, in the final cutting pattern, by at most two cuts. We solve the C_2TDC problem using an exact algorithm that is mainly based on a bottom-up strategy. We introduce new lower and upper bounds and propose new strategies that eliminate several duplicate patterns. We evaluate the performance of the proposed exact algorithm on problem instances extracted from the literature and compare it to the performance of an existing exact algorithm.