Fast approximation schemes for Boolean programming and scheduling problems related to positive convex Half-Product

We address a version of the Half-Product Problem and its restricted variant with a linear knapsack constraint. For these minimization problems of Boolean programming, we focus on the development of fully polynomial-time approximation schemes with running times that depend quadratically on the number of variables. Applications to various single machine scheduling problems are reported: minimizing the total weighted flow time with controllable processing times, minimizing the makespan with controllable release dates, minimizing the total weighted flow time for two models of scheduling with rejection

A fast FPTAS for single machine scheduling problem of minimizing total weighted earliness and tardiness about a large common due date

We address the single machine scheduling problem to minimize the total weighted earliness and tardiness about a nonrestrictive common due date. This is a basic problem with applications to the just-in-time manufacturing. The problem is linked to a Boolean programming problem with a quadratic objective function, known as the half-product. An approach to developing a fast fully polynomial-time approximation scheme (FPTAS) for the problem is identified and implemented. The running time matches the best known running time for an FPTAS for minimizing a half-product with no additive constan

Approximation schemes for non-separable non-linear Boolean programming problems under nested knapsack constraints

We consider a fairly general model of “take-or-leave”decision-making. Given a number of items of a particular weight, the decision-maker either takes (accepts) an item or leaves (rejects) it. We design fully polynomial-time approximation schemes (FPTASs) for optimization of a non-separable non-linear function which depends on which items are taken and which are left. The weights of the taken items are subject to nested constraints. There is a noticeable lack of approximation results on integer programming problems with non-separable functions. Most of the known positive results address special forms of quadratic functions, and in order to obtain the corresponding approximation algorithms and schemes considerable technical difficulties have to be overcome. We demonstrate how for the problem under consideration and its modifications FPTASs can be designed by using (i) the geometric rounding techniques, and (ii) methods of K -approximation sets and functions. While the latter approach leads to a faster scheme, the running times of the of both algorithms compare favorably with known analogues for less general problems

Approximation schemes for scheduling on a single machine subject to cumulative deterioration and maintenance

We consider a scheduling problem on a single machine to minimize the makespan. The processing conditions are subject to cumulative deterioration, but can be restored by a single maintenance. We link the problem to the Subset-sum problem (if the duration of maintenance is constant) and to the Half-Product Problem (if the duration of maintenance depends on its start time). For both versions of the problem, we adapt the existing fully polynomial-time approximation schemes to our problems by handling the additive constants

Single machine scheduling with a generalized job-dependent cumulative effect

We consider a single machine scheduling problem with changing processing times. The processing conditions are subject to a general cumulative effect, in which the processing time of a job depends on the sum of certain parameters associated with previously scheduled jobs. In previous papers, these parameters are assumed to be equal to the normal processing times of jobs, which seriously limits the practical application of this model. We further generalize this model by allowing every job to respond differently to these cumulative effects. For the introduced model, we solve the problem of minimizing the makespan, with and without precedence constraints. For the problem without precedence constraints, we also consider a situation in which a maintenance activity is included in the schedule, which can improve the processing conditions of the machine, not necessarily to its original state. The resulting problem is reformulated as a variant of a Boolean programming problem with a quadratic objective, known as a half-product, which allows us to develop a fully polynomial-time approximation scheme with the best possible running time