An important issue in multiobjective optimization is the quantitative comparison of the performance of different algorithms. In the case of multiobjective evolutionary algorithms, the outcome is usually an approximation of the Pareto-optimal set, which is denoted as an approximation set, and therefore the question arises of how to evaluate the quality of approximation sets. Most popular are methods that assign each approximation set a vector of real numbers that reflect different aspects of the quality. Sometimes, pairs of approximation sets are also considered. In this study, we provide a rigorous analysis of the limitations underlying this type of quality assessment. To this end, a mathematical framework is developed which allows one to classify and discuss existing techniques.

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Performance assessment of multiobjective optimizers: an analysis and review

Today, a variety of heuristic approaches are available to the operations research practitioner. One methodology that has a strong intuitive appeal, a prominent empirical track record, and is trivial to efficiently implement on parallel processors is GRASP (Greedy Randomized Adaptive Search Procedures). GRASP is an iterative randomized sampling technique in which each iteration provides a solution to the problem at hand. The incumbent solution over all GRASP iterations is kept as the final result. There are two phases within each GRASP iteration: the first intelligently constructs an initial solution via an adaptive randomized greedy function; the second applies a local search procedure to the constructed solution in hope of finding an improvement. In this paper, we define the various components comprising a GRASP and demonstrate, step by step, how to develop such heuristics for combinatorial optimization problems. Intuitive justifications for the observed empirical behavior of the methodology are discussed. The paper concludes with a brief literature review of GRASP implementations and mentions two industrial applications.

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Greedy Randomized Adaptive Search Procedures

Resource-constrained project scheduling: Notation, classification, models, and methods

Greedy Randomized Adaptive Search Procedures.

We consider the problem of scheduling n jobs to minimize the total earliness and tardiness penalty. We review the literature on this topic, providing a framework to show how results have been generalized starting with a basic model that contains symmetric penalties, one machine and a common due date. To this base we add such features as parallel machines, complex penalty functions and distinct due dates. We also consolidate many of the existing results by proving general forms of two key properties of earliness/tardiness models.

Sequencing with Earliness and Tardiness Penalties: A Review

The discrete time-cost tradeoff problem revisited

This note addresses the discrete version of the well-known time-cost tradeoff problem for project networks, which has been studied previously in the standard project management literature as well as in the related literature on Decision-CPM. All the algorithms proposed thus far for the solution of the general problem exhibit exponential worst-case complexity, with the notable exception of the pseudo-polynomial dynamic program due to Hindelang and Muth. We first demonstrate that this algorithm is flawed, and that when we correct it, it no longer remains pseudo-polynomial. Continuing on in the main result of the note, we show that this is not at all surprising, since the problem is strongly NP-hard. Finally, we discuss the complexities of various network structures and validate an old conjecture that certain structures are necessarily more difficult to solve.

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Complexity of the Discrete Time-Cost Tradeoff Problem for Project Networks

We discuss a single-machine scheduling problem where the objective is to minimize the variance of job completion times. To date, the problem has not been solved in polynomial time. This paper presents a dynamic programming algorithm that is pseudopolynomial in complexity. We also propose a fully polynomial approximation scheme and derive a lower bound that is useful in its implementation. Furthermore, we show that the dynamic programming solution is easy to extend to a bicriteria version of the problem in which it is desired to simultaneously minimize the mean completion time.

On the minimization of completion time variance with a bicriteria extension

This paper describes an integer programming model that we believe has advantages over other models for determining faculty teaching assignments. Unlike previously proposed models, in this model the decision variables represent the assignment of complete teaching schedules rather than courses to faculty members. The paper compares the proposed model with previous models for application in an academic department.

A mathematical programming model for faculty course assignments

This paper examines the problem of quoting a common delivery time for several orders. The decision maker responsible for the quote must trade off the cost associated with a long delivery time against the cost of tardy orders that might be induced by a short delivery time. First, an aggregate cost model is investigated where it is assumed that some cost is incurred if the quoted delivery time exceeds a given threshold and the cost of order tardiness is dependent on the order. Both exact and heuristic solution procedures are proposed for this model. Finally, for the situation where the costs cannot be quantified, a bicriteria formulation of the problem is pursued and the set of nondominated solutions is identified.

Charles E. Wells

Papers

The discrete time-cost tradeoff problem revisited

Complexity of the Discrete Time-Cost Tradeoff Problem for Project Networks

On the minimization of completion time variance with a bicriteria extension

A mathematical programming model for faculty course assignments

Optimal Delivery Time Quotation and Order Sequencing