Knapsack problem with objective value gaps

International audienceWe study a 0-1 knapsack problem, in which the objective value is forbidden to take some values. We call gaps related forbidden intervals. The problem is NP-hard and pseudo-polynomially solvable independently on the measure of gaps. If the gaps are large, then the problem is polynomially non-approximable. A non-trivial special case with respect to the approximate solution appears when the gaps are small and polynomially close to zero. For this case, two fully polynomial time approximation schemes are proposed. The results can be extended for the constrained longest path problem and other combinatorial problems

Batch Scheduling of Deteriorating Products

In this paper we consider the problem of scheduling N jobs on a single machine, where the jobs are processed in batches and the processing time of each job is a simple linear increasing function depending on job’s waiting time, which is the time between the start of the processing of the batch to which the job belongs and the start of the processing of the job. Each batch starts from the setup time S. Jobs which are assigned to the batch are being prepared for the processing during time S0 < S. After this preparation they are ready to be processed one by one. The non-negative number bi is associated with job i. The processing time of the i-th job is equal to bi(si − (sib + S0)), where sib and si are the starting time of the b-th batch to which the i-th job belongs and the starting time of this job, respectively. The objective is to minimize the completion time of the last job. We show that the problem is NP-hard. After that we present an O(N) time algorithm solving the problem optimally for the case bi = b. We further present an O(N2) time approximation algorithm with a performance guarantee 2

A Batching Machine Model for Lot Scheduling on a Single Machine

A recently introduced lot scheduling problem is considered. It is to find a partition of jobs of n orders into lots and to sequence these lots on a single machine so that the total average completion time of the orders is minimized. A simple O(n log n) time algorithm is presented for this problem in the literature, with a relatively sophisticated proof of its optimality. We show that modeling this problem as a classic batching machine problem makes its optimal solution obvious

Multi-product lot-sizing and sequencing on an imperfect single machine

http://www.emse.fr/spip/IMG/pdf/RR_2007-500-009.pdfWe study a problem of lot-sizing and sequencing several discrete products on a single machine. A sequence dependent setup time is required between the lots of di erent products. The machine is imperfect in the sense that it can produce defective items, and furthermore, it can break down. The number of the defective items of each product is given as an integer valued non-decreasing function of the manufactured quantity of this product, and the total machine breakdown time is given as a real valued non-decreasing function of the manufactured quantities of all the products. Two problem settings are considered. In the rst setting, the objective is to minimize the completion time of the last item, provided that all the product demands for the good quality items are satis ed. In the second setting, the objective is to minimize the total cost of the demand dissatisfaction, provided that a given upper bound on the completion time of the last item has been satis ed. We derive properties of the optimal solutions, NP-hardness proofs for the general cases, and polynomial exact and approximation algorithms for the case, in which the number of the defective items is given by rounding down a linear function of the manufactured quantity. This case is known as the \fraction defective" case in the quality control literature

Combinatorial design of a minimum cost transfer line with parallel operations at workstations

http://www.emse.fr/spip/IMG/pdf/RR_2007-500-010.pdfWe study a problem related to the optimal design of a transfer line. The line is to be used for a mass production of a single machine component, which requires a given set of operations to be performed. The line is a sequence of workstations, which number has to be decided, and each workstation has to be assigned a number of processing modules (blocks). A set of available blocks is given. There is the same cost associated with each workstation and there are di erent costs associated with the blocks. The problem is to design a minimum cost transfer line. The speci city of the problem is that all operations of the blocks assigned to the same workstation are performed in parallel so that the processing time of the machine component on the workstation is determined by the longest operation of this workstation, and the transfer line cycle time is determined by the longest operation of all the operations. Furthermore, there are inclusion, exclusion and precedence relations that restrict the assignment of the blocks and operations to the same workstation and the processing order of the operations on the transfer line. We suggest two combinatorial approaches for solving this problem, which are theoretically e cient if the number of blocks assigned to the same workstation and the number of workstations are upper bounded by given constants. The rst approach is to combinatorially enumerate all feasible solutions, and the second approach is to reduce the problem to the well studied maximum weight clique problem