Robinson Hoto

The one dimensional Compartmentalised Knapsack Problem: A case study

The Compartmentalised Knapsack Problem (CKP) is similar to the ordinary Knapsack Problem except that items to be packed belong to separate classes, and items can only be packed, in knapsack compartments, amongst items in their own class. This paper addresses a case study in the cutting of steel rolls in which the CKP arises. The rolls are cut in two-phases: the first phase produces sub-rolls (compartments) which are, after reducing the thickness, cut in a second phase to produce ribbons (a class consists of ordered items with the same thickness). Finally, two methods of solving CKP are presented, and these are used to generate columns in the classical linear optimisation model of Gilmore and Gomory. Results of computational experiments are presented.

The constrained compartmentalized knapsack problem: mathematical models and solution methods

The constrained compartmentalized knapsack problem can be seen as an extension of the constrained knapsack problem. However, the items are grouped into different classes so that the overall knapsack has to be divided into compartments, and each compartment is loaded with items from the same class. Moreover, building a compartment incurs a fixed cost and a fixed loss of the capacity in the original knapsack, and the compartments are lower and upper bounded. The objective is to maximize the total value of the items loaded in the overall knapsack minus the cost of the compartments. This problem has been formulated as an integer non-linear program, and in this paper, we reformulate the non-linear model as an integer linear master problem with a large number of variables. Some heuristics based on the solution of the restricted master problem are investigated. A new and more compact integer linear model is also presented, which can be solved by a branch-and-bound commercial solver that found most of the optimal solutions for the constrained compartmentalized knapsack problem. On the other hand, heuristics provide good solutions with low computational effort.

A Study of the Compartimentalized Knapsack Problem with Additional Restrictions

The Compartimentalized Knapsack Problem has been reported in the literature to generate cutting patterns of problems in two stages. The problem has constrained case, which are considered limits on the number of compartments and items in the knapsack. An exact algorithm that involves the resolution of various knapsacks and heuristics based on upper bound has already been developed. In this paper we present the problem with mathematical models and propose new strategies for resolving the constrained case.

Um problema de corte com padrões compartimentados

In this paper we will present the application of the Compartmented Knapsack Problem (CKP) in the Cut Problem of Steel Rolls (CPSR), that it is a problem of cut in two stages with restrictions special of grouping of items. The CKP consists of constructing compartments of unknown capacities in a knapsack of known capacity, in view of that items of interest is grouped in subgroups, in mode that, items of a grouping cannot be matched with items of another one. To understand the CKP more good it admits that the knapsack of a alpinist must be composite for an ideal number of compartments with items of four categories (remedies, foods, tools, clothes), however, items of distinct categories cannot be matched to form one same compartment, in addition, is unknown the ideal capacities of each compartment of the knapsack.

Reducing the Setup of a Tubettes Machine

This manuscript deals with the manufacturing process of tubettes (tubes made by gluing strips of paper), whose preparation of tubettes machine occupies a considerable portion of production time. The paper strips are packed in paper reels and some of them may be used between one and making another tubettes. Two mathematical models for the minimization of change reels are shown. The results obtained by simulations with the solver Xpress-MP were better than those used by the company, with reductions of up to 37% in the number of changes.

MINIMIZING THE PREPARATION TIME OF A TUBES MACHINE: EXACT SOLUTION AND HEURISTICS

In this paper we optimize the preparation time of a tubes machine. Tubes are hard tubes made by gluing strips of paper that are packed in paper reels, and some of them may be reused between the production of one and another tube. We present a mathematical model for the minimization of changing reels and movements and also implementations for the heuristics Nearest Neighbor, an improvement of a nearest neighbor (Best Nearest Neighbor), refinements of the Best Nearest Neighbor heuristic and a heuristic of permutation called Best Configuration using the IDE (integrated development environment) WxDev C++. The results obtained by simulations improve the one used by the company.

New solutions to the constrained compartmentalised knapsack problem

The compartmentalised knapsack problem has been described in the literature as generating patterns of cutting problems in two phases. An exact algorithm, which involves the solution of several knapsacks, and another one, based on upper bounds, have already been developed. In the constrained case of a compartmentalised knapsack, in which bounds in the number of compartments and items are required, few results are known. In this paper, we present a new heuristic to solve the constrained case of the compartmentalised knapsack problem and make comparisons with the best heuristic registered in the literature.