Improved algorithm for a generalized on-line scheduling problem on identical machines

This paper considers the problem of on-line scheduling a list of independent jobs in which each job has an arbitrary release time on m parallel identical machines. In this problem, jobs arrive in form of order before its release time and decisions have to be made whenever an order is placed and the orders arrive according to any sequence. A heuristic algorithm, NMLS, better than MLS is given for any m ? 2. The competitive ratio is improved from 2.93920 to 2.78436.     

Haplotyping populations by pure parsimony based on compatible genotypes and greedy heuristics

The population haplotype inference problem based on the pure parsimony criterion (HIPP) infers an m × n genotype matrix for a population by a 2m × n haplotype matrix with the minimum number of distinct haplotypes. Previous integer programming based HIPP solution methods are time-consuming, and their practical effectiveness remains unevaluated. On the other hand, previous heuristic HIPP algorithms are efficient, but their theoretical effectiveness in terms of optimality gaps has not been evaluated, either. We propose two new heuristic HIPP algorithms (MGP and GHI) and conduct more complete computational experiments. In particular, MGP exploits the compatible relations among genotypes to solve a reduced integer linear programming problem so that a solution of good quality can be obtained very quickly; GHI exploits a weight mechanism to selects better candidate haplotypes in a greedy fashion. The computational results show that our proposed algorithms are efficient and effective, especially for solving cases with larger recombination rates.     

The p-median problem under uncertainty

Consider the need to currently locate p facilities but it is possible that up to q additional facilities will have to be located in the future. There are known probabilities that 0 ? r ? q facilities will need to be located. The p-median problem under uncertainty is to find the location of p facilities such that the expected value of the objective function in the future is minimized. The problem is formulated on a graph, properties of it are proven, an integer programming formulation is constructed, and heuristic algorithms are suggested for its solution. The heuristic algorithms are modified to reduce the run time by about two orders of magnitude with minimal effect on the quality of the solution. Optimal solutions for many problems are found effectively by CPLEX. Computational results using the heuristic algorithms are presented.     

Enumerating K best paths in length order in DAGs

We address the problem of finding the K best paths connecting a given pair of nodes in a directed acyclic graph (DAG) with arbitrary lengths. One of the main results in this paper is the proof that a tree representing the kth shortest path is obtained by an arc exchange in one of the previous (k − 1) trees (each of which contains a previous best path). An O(m + K(n + log K)) time and O(K + m) space algorithm is designed to explicitly determine the K shortest paths in a DAG with n nodes and m arcs. The algorithm runs in O(m + Kn) time using O(K + m) space in DAGs with integer length arcs. Empirical results confirming the superior performance of the algorithm to others found in the literature for randomly generated graphs are reported.     

A bicriteria flowshop scheduling with a learning effect

In many situations, a worker’s ability improves as a result of repeating the same or similar tasks; this phenomenon is known as the learning effect. In this paper the learning effect is considered in a two-machine flowshop. The objective is to find a sequence that minimizes a weighted sum of total completion time and makespan. Total completion time and makespan are widely used performance measures in scheduling literature. To solve this scheduling problem, an integer programming model with n² + 6n variables and 7n constraints where n is the number of jobs is formulated. Because of the lengthy computing time and high computing complexity of the integer programming model, the problem with up to 30 jobs can be solved. A heuristic algorithm and a tabu search based heuristic algorithm are presented to solve large size problems. Experimental results show that the proposed heuristic methods can solve this problem with up to 300 jobs rapidly. According to the best of our knowledge, no work exists on the bicriteria flowshop with a learning effect.     

Preemptive multiprocessor order scheduling to minimize total weighted flowtime

Consider m identical machines in parallel, each of which can produce k different product types. There is no setup cost when the machines switch from producing one product type to another. There are n orders each of which requests various quantities of the different product types. All orders are available for processing at time t = 0, and preemption is allowed. Order i has a weight w_i and its completion time is the time when its last requested product type finishes. Our goal is to find a preemptive schedule such that the total weighted completion time ∑_wiCi$\sum w_i{C}_i$ is minimized. We show that this problem is NP-hard even when all jobs have identical weights and there are only two machines. Motivated by the computational complexity of the problem, we propose a simple heuristic and show that it obeys a worst-case bound of 2 − 1/m. Finally, empirical studies show that our heuristic performs very well when compared with a lower bound of the optimal cost.     

A new heuristic for open shop total completion time problem

The m  -machine open shop problem to minimize the sum of the completion times is investigated in our paper. First, a heuristic algorithm, Shortest Processing Time Block, is presented to deal with problem _Om|n=km|∑_Cj$O_m|n=\mathit{km}|\sum C_j$, where k   is positive integer. And then, the heuristic is extended to solve the general problem _Om‖∑_Cj$O_m\Vert \sum C_j$. It is proved that the heuristic is asymptotically optimal when the job number goes to infinity. At the end of this paper, numerical experimentation results show the effectiveness of the heuristic algorithm.     

The generating function of a family of the sequences in terms of the continuant

Let a₀, a₁, … , a_r−1 be positive numbers and define a sequence {q_m}, with initial conditions q₀ = 0 and q₁ = 1, and for all m ? 2, q_m = a_tq_m−1 + q_m−2 where m ≡ t(mod r). For r = 2, the author called the sequence {q_m} as the generalized Fibonacci sequences and studied it in [1]. But, it remains open to find a closed form of the generating function for general {q_m}. In this paper, we solve this open problem, that is, we find a closed form of the generating function for {q_m}in terms of the continuant.     

Minimum flow problem on network flows with time-varying bounds

In this paper, we consider the minimum flow problem on network flows in which the lower arc capacities vary with time. We will show that this problem for set {0, 1, … , T} of time points can be solved by at most n minimum flow computations, by combining of preflow-pull algorithm and reoptimization techniques (no matter how many values of T are given). Running time of the presented algorithm is O(n²m).     

On-line scheduling to minimize max flow time: an optimal preemptive algorithm

We investigate the maximum flow time minimization problem of on-line scheduling jobs on m identical parallel machines. When preemption is allowed, we derive an optimal algorithm with competitive ratio 2-1/m. When preemption is not allowed and m=2, we show that the First In First Out heuristic achieves the best possible competitive ratio.