Anup K. Sen

Single machine weighted earliness–tardiness penalty problem with a common due date ☆

Abstract In this paper, we have considered a class of single machine job scheduling problems where the objective is to minimize the weighted sum of earliness–tardiness penalties of jobs. The weights are job-independent but they depend on whether a job is early or tardy. The restricted version of the problem where the common due date is smaller than a critical value, is known to be NP-complete. While dynamic programming formulation runs out of memory for large problem instances, depth-first branch-and-bound formulation runs slow for large problems since it uses a tree search space. In this paper, we have suggested an algorithm to optimally solve large instances of the restricted version of the problem. The algorithm uses a graph search space. Unlike dynamic programming, the algorithm can output optimal solutions even when available memory is limited. It has been found to run faster than dynamic programming and depth-first branch-and-bound formulations and can solve much larger instances of the problem in reasonable time. New upper and lower bounds have been proposed and used. Experimental findings are given in detail. Scope and purpose A class of single machine problems arising out of scheduling jobs in JIT environment has been considered in this paper. The objective is to minimize the total weighted earliness–tardiness penalties of jobs. In this paper, we have presented a new algorithm and conducted extensive empirical runs to show that the new algorithm performs much better than the existing approaches in solving large instances of the problem.

Graph search methods for non-order-preserving evaluation functions: applications to job sequencing problems

Abstract Graph search with A ∗ is frequently faster than tree search. But A ∗ graph search operates correctly only when the evaluation function is order-preserving. In the non-orderpreserving case, no paths can be discarded and the entire explicit graph must be stored in memory. Such situations arise in one-machine minimum penalty job sequencing problems when setup times are sequence dependent. GREC, the unlimited memory version of a memory-constrained search algorithm of the authors called MREC, has a clear advantage over A ∗ in that it is able to find optimal solutions to such problems. At the same time, it is as efficient as A ∗ in solving graph search problems with order-preserving evaluation functions. Experimental results indicate that in the non-order-preserving case, GREC is faster than both best-first and depth-first tree search, and can solve problem instances of larger size than best-first tree search.

An improved precedence rule for single machine sequencing problems with quadratic penalty

Abstract This article introduces a new heuristic for the single machine job sequencing problem where the objective is to minimize the weighted sum of quadratic completion times of jobs. An improved precedence constraint among non-adjacent jobs has been conjectured. The conjecture appears to be difficult to prove though extensive empirical runs failed to generate even a single counter example. This leads to an interesting observation: if the conjecture is used to develop a heuristic, it will yield optimal solutions in all instances of the extensive empirical runs performed by us. We expect that the heuristic would produce optimal solutions for other instances too. The conjecture, therefore, raises an open problem for further research in this domain.

On Detecting Data Flow Errors in Workflows

When designing a business workflow, it is customary practice to create the control flow structure first and to ensure its correctness. Information about the flow of data is introduced subsequently into the workflow and its correctness is independently verified. Improper specification of data requirements of tasks and XOR splits can cause problems such as wrong branching at XOR splits and the failure of tasks to execute. Here we present a graph traversal algorithm called GTforDF for detecting data flow errors in both nested and unstructured workflows, and illustrate its operation on realistic examples. Two of these have interconnected loops and are free of control flow errors, and the third one is an unstructured loop-free workflow. Our approach extends and generalizes data flow verification methods that have been recently proposed. It also makes use of the concept of corresponding pairs lately introduced in control flow verification. It thus has the potential for development into a unified algorithmic procedure for the concurrent detection of control flow and data flow errors. The correctness of the algorithm has been proved theoretically. It has also been tested experimentally on many examples.

Searching networks with unrestricted edge costs

Best-first and depth-first heuristic search algorithms often assume underlying search graphs with only nonnegative edge costs and attempt to optimize simple objective functions. Applicability of these algorithms to graphs with both positive and negative edge costs is not completely studied. In the paper, two new problems are identified: one in computational geometry and the other in the layout design of very large scale integrated (VLSI) circuits. The former problem relates to a weight-balanced bipartitioning of a given set of points in a plane. The goal of the second problem is to find an area-balanced staircase path in a VLSI floorplan. Formulations of these problems lead to an interesting directed acyclic search graph with positive, zero and negative edge costs and an objective function of general nature. These problems are NP-hard. To solve such general problems optimally, search schemes are proposed. Experimental results reveal the efficacy and versatility of the proposed schemes, the depth-first scheme being the better choice. It is shown that the classical number-partitioning problem can also be formulated in this framework.

Detecting Data Flow Errors in Workflows: A Systematic Graph Traversal Approach

When designing a workflow, it is customary practice to create the control flow structure first and to ensure its correctness. Information about the flow of data is introduced subsequently into the workflow and its correctness is independently verified. Improper specification of data requirements of tasks and XOR splits can cause problems such as wrong branching at XOR splits and the failure of tasks to execute. Here we present a graph traversal algorithm called GTforDF for detecting data flow errors in a workflow that is free of control flow errors, and illustrate its operation on two realistic workflows with interconnected loops. Our approach extends and generalizes data flow verification methods that have been recently proposed. It also makes use of the concept of corresponding pairs lately introduced in control-flow verification. It thus has the potential for development into a unified algorithmic procedure for the concurrent detection of control flow and data flow errors.

Application of graph search and genetic algorithms for the single machine scheduling problem with sequence-dependent setup times and quadratic penalty function of completion times

In this paper, we consider the single machine scheduling problem with quadratic penalties and sequence-dependent (QPSD) setup times. QPSD is known to be NP-Hard. Only a few exact approaches, and to the best of our knowledge, no approximate approaches, have been reported in the literature so far. This paper discusses exact and approximate approaches for solving the problem, and presents empirical findings. We make use of a graph search algorithm, Memory-Based Depth-First Branch-and-Bound (MDFBB), and present an algorithm, QPSD_MDFBB that can optimally solve QPSD, and advances the state of the art for finding exact solutions. For finding approximate solutions to large problem instances, we make use of the idea of greedy stochastic search, and present a greedy stochastic algorithm, QPSD_GSA that provides moderately good solutions very rapidly even for large problems. The major contribution of the current paper is to apply QPSD_GSA to generate a subset of the starting solutions for a new genetic algorithm, QPSD_GEN, which is shown to provide near-optimal solutions very quickly. Owing to its polynomial running time, QPSD_GEN can be used for much larger instances than QPSD_MDFBB can handle. Experimental results have been provided to demonstrate the performances of these algorithms.

Searching graphs with A*: applications to job sequencing

The best-first search algorithm A* allows search graphs that are trees, directed acyclic graphs or directed graphs with cycles. In real life applications of A* the search graph is generally implemented as a tree. It is shown here that for certain well known one-machine job sequencing problems that arise in job shops, graph search is much faster than best-first tree search when problem instances are of small and medium size. Moreover, graph search uses less memory and so is able to solve larger problems. Depth-first search needs little memory, and is therefore capable in principle of solving problems of arbitrary size, but is slower than graph search by orders of magnitude for the examples that were studied.

Greedy by Chance - Stochastic Greedy Algorithms

For many complex combinatorial optimization problems, obtaining good solutions quickly is of value either by itself or as part of an exact algorithm. Greedy algorithms to obtain such solutions are known for many problems. In this paper we present stochastic greedy algorithms which are perturbed versions of standard greedy algorithms, and report on experiments using learned and standard probability distributions conducted on knapsack problems and single machine sequencing problems. The results indicate that the approach produces solutions significantly closer to optimal than the standard greedy approach, and runs quite fast. It can thus be seen in the space of approximate algorithms as falling between the very quick greedy approaches and the relatively slower soft computing approaches like genetic algorithms and simulated annealing.

Average-case analysis of heuristic search in tree-like networks

A search graph has the form of an m-ary tree with bi-directional arcs of unit cost. There is a goal node at a distance N from the root, and there may be other goal nodes at distances ≥ N from the root. It is assumed that the heuristic estimates of nongoal nodes, after being appropriately normalized, are independent and identically distributed random variables. The heuristic is not required to be admissible. Under what conditions is the expected number of node expansions E(Z) polynomial in N? Earlier efforts by Pearl and others at answering this question have considered search trees with only one goal node. An attempt is made here to develop a general and unified method of analysis applicable to situations with more than one goal node. It is shown that, for most probability distributions on the heuristic estimates, E(Z) is exponential in N; the one major exception being the case when the number of goal nodes is polynomial in N and the normalizing function for the error is logarithmic. Pearl’s contention that the average-case analysis of weighted heuristic search is not too attractive is also verified. It is hoped that the general approach described here will encourage similar studies on search graphs other than trees.