An integer linear optimization model to the compartmentalized knapsack problem

Inarejos, Osvaldo; Hoto, Robinson; Maculan, Nelson

doi:10.1111/itor.12490

Cited by 5 publications

(21 citation statements)

References 15 publications

(24 reference statements)

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“…In the following sections, we use the mixed‐integer LP model of Inarejos et al. (), and adjust the value of data coefficients, lift coefficients, and other IP tools to obtain a stronger model, with a reduced domain, defined as follows. Definition Consider equivalent models IP and IP′ with LP relaxations with sets X and

X^{'}

, respectively.…”

Section: A Strong Linear Model Of the Compartmentalized Knapsack Problemmentioning

confidence: 99%

“…Inarejos et al. () present a linear model (LM) for the CKPR (LMCKPR) and proves its linearity, in addition to exposing a new heuristic for the CKPR, using the classic model of the problem, called the decreasing classes heuristic , and claiming it to be more competitive in relation to the heuristic of the W capacities of Marques and Arenales (). The linearity of LMCKPR is an important contribution of Inarejos et al.…”

Section: Introductionmentioning

confidence: 99%

“…The linearity of LMCKPR is an important contribution of Inarejos et al. (), because it makes the model amenable to solution by mixed‐integer linear programming (LP) tools.…”

Section: Introductionmentioning

confidence: 99%

“…For modeling the CKPR, in order to obtain an LM, Inarejos et al. () discretize the compartments and introduce a limit to the number of compartments that can be created for each class

k = 1, \dots, q

, calculated as

p_{k} = min false{F_{1}, ⌊ L / L_{m i n}^{k} ⌋ false}

. To facilitate the handling of all problem information, local indexing of the items of each class from 1 to

false| N_{k} false|

and the compartments for each class from 1 to

p_{k}

is used.…”

Section: Introductionmentioning

confidence: 99%

“…The decision variables

a_{i j}^{k}

represent the number of items i in the

j th

compartment of the

k th

class,

\forall i \in N_{k}, j = 1, \dots, p_{k}, k = 1, \dots, q

. The LM for the CKPR (see Inarejos et al., ) is the following:

0true maximize: \sum_{k = 1}^{q} \sum_{i \in N_{k}} \sum_{j = 1}^{p_{k}} u_{i}^{k} a_{i j}^{k},

0true subject to: \sum_{k = 1}^{q} \sum_{i \in N_{k}} \sum_{j = 1}^{p_{k}} l_{i}^{k} a_{i j}^{k} \leq L,

0true δ_{j}^{k} L_{m i n}^{k} \leq \sum_{i \in N_{k}} l_{i}^{k} a_{i j}^{k} \leq δ_{j}^{k} L_{m a x}^{k}, \forall j = 1, \dots, p_{k}, k = 1, \dots, q,

0true \sum_{j = 1}^{p_{k}} a_{i j}^{k} \leq d_{i}^{k}, i \in N_{k}, k = 1, \dots, q, ...

…”

Section: Introductionmentioning

confidence: 99%

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A strong integer linear optimization model to the compartmentalized knapsack problem

Quiroga‐Orozco

Carvalho

Hoto

2019

Int Trans Operational Res

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The compartmentalized knapsack problem (CKP) is a relatively new type of problem with a wide application in industrial processes, arising, for instance, in the case of cutting steel coils in two phases in the metallurgical industry. In the literature, there are two mathematical formulations for the CKP: a classical formulation, which is a nonlinear integer programming (IP) model, and a recent (linear) IP formulation, obtained by discretizing the compartments that can be built for each class of items; the latter is an important contribution, because it makes the problem amenable to solution by mixed‐integer linear programming tools. Combinatorial enumeration algorithms and several pseudo‐polynomial decomposition heuristics were also developed for the CKP. This paper presents a new model for the exact solution of the CKP, denoted as the strong integer linear model, derived from the (linear) IP formulation by strengthening data, reducing symmetry, and lifting, and also a new pseudo‐polynomial heuristic, the heuristic of the pk strong capacities. Computational experiments are presented with a large set of instances that show the advantage of the new approaches. The strong model solves the CKP exactly more than seven times faster, and the new heuristic is more efficient, presenting a good balance in the terms of effectiveness.

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X^{'}

, respectively.…”

Section: A Strong Linear Model Of the Compartmentalized Knapsack Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The linearity of LMCKPR is an important contribution of Inarejos et al. (), because it makes the model amenable to solution by mixed‐integer linear programming (LP) tools.…”

Section: Introductionmentioning

confidence: 99%

“…For modeling the CKPR, in order to obtain an LM, Inarejos et al. () discretize the compartments and introduce a limit to the number of compartments that can be created for each class