Group testing procedures with quantitative features and incomplete identification

We present a group testing model for items characterized by marker random variables. An item is defined to be good (defective) if its marker is below (above) a given threshold. The items can be tested in groups; the goal is to obtain a prespecified number of good items by testing them in optimally sized groups. Besides this group size, the controller has to select a threshold value for the group marker sums, and the target number of groups which by the tests are classified to consist only of good items. These decision variables have to be chosen so as to minimize a cost function, which is a linear combination of the expected number of group tests and an expected penalty for missing the desired number of good items, subject to constraints on the probabilities of misclassifications. We treat two models of this kind: the first one is based on an infinite population size, whereas the second one deals with the case of a finite number of available items. All performance measures are derived in closed form; approximations are also given. Furthermore, we prove monotonicity properties of the components of the objective function and of the constraints. In several examples, we study (i) the dependence of the cost function on the decision variables and (ii) the dependence of the optimal values of the decision variables (group size, group marker threshold, and stopping rule for groups classified as clean) and of the target functionals (optimal expected number of tests, optimal expected penalty, and minimal expected cost) on the system parameters.© 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Dynamic routing of prioritized warranty repairs

Recent years have seen a strong trend toward outsourcing warranty repair services to outside vendors. In this article we consider the problem of dynamically routing warranty repairs to service vendors when warranties have priority levels. Each time an item under warranty fails, it is sent to one of the vendors for repair. Items covered by higher priority warranty receive higher priority in repair service. The manufacturer pays a fixed fee per repair and incurs a linear holding cost while an item is undergoing or waiting for repair. The objective is to minimize the manufacturer's long‐run average cost. Because of the complexity of the problem, it is very unlikely that there exist tractable ways to find the optimal routing strategies. Therefore, we propose five heuristic routing procedures that are applicable to real‐life problems. We evaluate the heuristics using simulation. The simulation results show that the index‐based “generalized join the shortest queue” policy, which applies a single policy improvement step to an initial state‐independent policy, performs the best among all five heuristics. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

A competitive genetic algorithm for resource-constrained project scheduling

In this paper we consider the resource-constrained project scheduling problem (RCPSP) with makespan minimization as objective. We propose a new genetic algorithm approach to solve this problem. Subsequently, we compare it to two genetic algorithm concepts from the literature. While our approach makes use of a permutation based genetic encoding that contains problem-specific knowledge, the other two procedures employ a priority value based and a priority rule based representation, respectively. Then we present the results of our thorough computational study for which standard sets of project instances have been used. The outcome reveals that our procedure is the most promising genetic algorithm to solve the RCPSP. Finally, we show that our genetic algorithm yields better results than several heuristic procedures presented in the literature. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 733–750, 1998     

Scheduling maintenance and semiresumable jobs on a single machine

The majority of scheduling literature assumes that the machines are available at all times. In this paper, we study single machine scheduling problems where the machine maintenance must be performed within certain intervals and hence the machine is not available during the maintenance periods. We also assume that if a job is not processed to completion before the machine is stopped for maintenance, an additional setup is necessary when the processing is resumed. Our purpose is to schedule the maintenance and jobs to minimize some performance measures. The objective functions that we consider are minimizing the total weighted job completion times and minimizing the maximum lateness. In both cases, maintenance must be performed within a fixed period T, and the time for the maintenance is a decision variable. In this paper, we study two scenarios concerning the planning horizon. First, we show that, when the planning horizon is long in relation to T, the problem with either objective function is NP-complete, and we present pseudopolynomial time dynamic programming algorithms for both objective functions. In the second scenario, the planning horizon is short in relation to T. However, part of the period T may have elapsed before we schedule any jobs in this planning horizon, and the remaining time before the maintenance is shorter than the current planning horizon. Hence we must schedule one maintenance in this planning horizon. We show that the problem of minimizing the total weighted completion times in this scenario is NP-complete, while the shortest processing time (SPT) rule and the earliest due date (EDD) rule are optimal for the total completion time problem and the maximum lateness problem respectively. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 845–863, 1999     

Quadratic bottleneck problems

We study the quadratic bottleneck problem (QBP) which generalizes several well‐studied optimization problems. A weak duality theorem is introduced along with a general purpose algorithm to solve QBP. An example is given which illustrates duality gap in the weak duality theorem. It is shown that the special case of QBP where feasible solutions are subsets of a finite set having the same cardinality is NP‐hard. Likewise the quadratic bottleneck spanning tree problem (QBST) is shown to be NP‐hard on a bipartite graph even if the cost function takes 0–1 values only. Two lower bounds for QBST are derived and compared. Efficient heuristic algorithms are presented for QBST along with computational results. When the cost function is decomposable, we show that QBP is solvable in polynomial time whenever an associated linear bottleneck problem can be solved in polynomial time. As a consequence, QBP with feasible solutions form spanning trees, s‐t paths, matchings, etc., of a graph are solvable in polynomial time with a decomposable cost function. We also show that QBP can be formulated as a quadratic minsum problem and establish some asymptotic results. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Minimizing sums and products of linear fractional functions over a polytope

In this paper, we develop efficient deterministic algorithms for globally minimizing the sum and the product of several linear fractional functions over a polytope. We will show that an elaborate implementation of an outer approximation algorithm applied to the master problem generated by a parametric transformation of the objective function serves as an efficient method for calculating global minima of these nonconvex minimization problems if the number of linear fractional terms in the objective function is less than four or five. It will be shown that the Charnes–Cooper transformation plays an essential role in solving these problems. Also a simple bounding technique using linear multiplicative programming techniques has remarkable effects on structured problems. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 583–596, 1999     

A squared-variable transformation approach to nonlinear programming optimality conditions

We show that the well-known necessary and sufficient conditions for a relative maximum of a nonlinear differentiable objective function with nonnegative variables constrained by nonlinear differentiable inequalities may be derived using the classical theory of equality constrained optimization problems with unrestricted variables. To do this we transform the original inequality-constrained problem to an equivalent equality-constrained problem by means of a well-known squared-variable transformation. Our major result is to show that second order conditions must be used to obtain the Kuhn-Tucker conditions by this approach. Our nonlinear programming results are motivated by the development of some well-known linear programming results by this approach.     

Scheduling with tool changes to minimize total completion time: A study of heuristics and their performance

The machine scheduling literature does not consider the issue of tool change. The parallel literature on tool management addresses this issue but assumes that the change is due only to part mix. In practice, however, a tool change is caused most frequently by tool wear. That is why we consider here the problem of scheduling a set of jobs on a single CNC machine where the cutting tool is subject to wear; our objective is to minimize the total completion time. We first describe the problem and discuss its peculiarities. After briefly reviewing available theoretical results, we then go on to provide a mixed 0–1 linear programming model for the exact solution of the problem; this is useful in solving problem instances with up to 20 jobs and has been used in our computational study. As our main contribution, we next propose a number of heuristic algorithms based on simple dispatch rules and generic search. We then discuss the results of a computational study where the performance of the various heuristics is tested; we note that the well‐known SPT rule remains good when the tool change time is small but deteriorates as this time increases and further that the proposed algorithms promise significant improvement over the SPT rule. © 2002 Wiley Periodicals, Inc. Naval Research Logistics, 2003     

Multiple subset sum with inclusive assignment set restrictions

In a traditional multiple subset sum problem (MSSP), there is a given set of items and a given set of bins (or knapsacks) with identical capacities. The objective is to select a subset of the items and pack them into the bins such that the total weight of the selected items is maximized. However, in many applications of the MSSP, the bins have assignment restrictions. In this article, we study the subset sum problem with inclusive assignment set restrictions, in which the assignment set of one item (i.e., the set of bins that the item may be assigned to) must be either a subset or a superset of the assignment set of another item. We develop an efficient 0.6492‐approximation algorithm and test its effectiveness via computational experiments. We also develop a polynomial time approximation scheme for this problem. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Minimizing the makespan in open‐shop scheduling problems with a convex resource consumption function

We consider open‐shop scheduling problems where operation‐processing times are a convex decreasing function of a common limited nonrenewable resource. The scheduler's objective is to determine the optimal job sequence on each machine and the optimal resource allocation for each operation in order to minimize the makespan. We prove that this problem is NP‐hard, but for the special case of the two‐machine problem we provide an efficient optimization algorithm. We also provide a fully polynomial approximation scheme for solving the preemptive case. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006     

Programming with linear fractional functionals

Charnes and Cooper [1] showed that a linear programming problem with a linear fractional objective function could be solved by solving at most two ordinary linear programming problems. In addition, they showed that where it is known a priori that the denominator of the objective function has a unique sign in the feasible region, only one problem need be solved. In the present note it is shown that if a finite solution to the problem exists, only one linear programming problem must be solved. This is because the denominator cannot have two different signs in the feasible region, except in ways which are not of practical importance.     

Scheduling parallel dedicated machines with the speeding‐up resource

We consider a problem of scheduling jobs on m parallel machines. The machines are dedicated, i.e., for each job the processing machine is known in advance. We mainly concentrate on the model in which at any time there is one unit of an additional resource. Any job may be assigned the resource and this reduces its processing time. A job that is given the resource uses it at each time of its processing. No two jobs are allowed to use the resource simultaneously. The objective is to minimize the makespan. We prove that the two‐machine problem is NP‐hard in the ordinary sense, describe a pseudopolynomial dynamic programming algorithm and convert it into an FPTAS. For the problem with an arbitrary number of machines we present an algorithm with a worst‐case ratio close to 3/2, and close to 3, if a job can be given several units of the resource. For the problem with a fixed number of machines we give a PTAS. Virtually all algorithms rely on a certain variant of the linear knapsack problem (maximization, minimization, multiple‐choice, bicriteria). © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

The fractional fixed-charge problem

Fractional fixed-charge problems arise in numerous applications, where the measure of economic performance is the time rate of earnings or profit (equivalent to an interest rate on capital investment). This paper treats the fractional objective function, after suitable transformation, as a linear parametric fixed-charge problem. It is proved, with wider generality than in the case of Hirsch and Dantzig, that some optimal solution to the generalized linear fixed-charge problem is an extreme point of the polyhedral set defined by the constraints. Furthermore, it is shown that the optimum of the generalized fractional fixed-charge problem is also a vertex of this set. The proof utilizes a suitable penalty function yielding an upper bound on the optimal value of the objective function; this is particularly useful when considering combinations of independent transportation-type networks. Finally, it is shown that the solution of a fractional fixed-charge problem is obtainable through that of a certain linear fixed-charge one.     

A finite algorithm for concave minimization over a polyhedron

We present a new algorithm for solving the problem of minimizing a nonseparable concave function over a polyhedron. The algorithm is of the branch-and-bound type. It finds a globally optimal extreme point solution for this problem in a finite number of steps. One of the major advantages of the algorithm is that the linear programming subproblems solved during the branch-and-bound search each have the same feasible region. We discuss this and other advantages and disadvantages of the algorithm. We also discuss some preliminary computational experience we have had with our computer code for implementing the algorithm. This computational experience involved solving several bilinear programming problems with the code.     

Multiple-facility loading under capacity-based economies of scope

Multiple-facility loading (MFL) involves the allocation of products among a set of finite-capacity facilities. Applications of MFL arise naturally in a variety of production scheduling environments. MFL models typically assume that capacity is consumed as a linear function of products assigned to a facility. Product similarities and differences, however, result in capacity-based economies or diseconomies of scope, and thus the effective capacity of the facility is often a (nonlinear) function of the set of tasks assigned to the facility. This article addresses the multiple-facility loading problem under capacity-based economies (and diseconomies) of scope (MFLS). We formulate MFLS as a nonlinear 0–1 mixed-integer programming problem, and we discuss some useful properties. MFLS generalizes many well-known combinatorial optimization problems, such as the capacitated facility location problem and the generalized assignment problem. We also define a tabu-search heuristic and a branch-and-bound algorithm for MFLS. The tabu-search heuristic alternates between two search phases, a regional search and a diversification search, and offers a novel approach to solution diversification. We also report computational experience with the procedures. In addition to demonstrating MFLS problem tractability, the computational results indicate that the heuristic is an effective tool for obtaining high-quality solutions to MFLS. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44: 229–256, 1997     

A branch‐and‐price approach for the stochastic generalized assignment problem

In this article, we address a stochastic generalized assignment machine scheduling problem in which the processing times of jobs are assumed to be random variables. We develop a branch‐and‐price (B&P) approach for solving this problem wherein the pricing problem is separable with respect to each machine, and has the structure of a multidimensional knapsack problem. In addition, we explore two other extensions of this method—one that utilizes a dual‐stabilization technique and another that incorporates an advanced‐start procedure to obtain an initial feasible solution. We compare the performance of these methods with that of the branch‐and‐cut (B&C) method within CPLEX. Our results show that all B&P‐based approaches perform better than the B&C method, with the best performance obtained for the B&P procedure that includes both the extensions aforementioned. We also utilize a Monte Carlo method within the B&P scheme, which affords the use of a small subset of scenarios at a time to estimate the “true” optimal objective function value. Our experimental investigation reveals that this approach readily yields solutions lying within 5% of optimality, while providing more than a 10‐fold savings in CPU times in comparison with the best of the other proposed B&P procedures. © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 131–143, 2014     

Vehicle routing problems with regular objective functions on a path

We investigate the problem of scheduling a fleet of vehicles to visit the customers located on a path to minimize some regular function of the visiting times of the customers. For the single‐vehicle problem, we prove that it is pseudopolynomially solvable for any minsum objective and polynomially solvable for any minmax objective. Also, we establish the NP‐hardness of minimizing the weighted number of tardy customers and the total weighted tardiness, and present polynomial algorithms for their special cases with a common due date. For the multivehicle problem involving n customers, we show that an optimal solution can be found by solving or O(n) single‐vehicle problems. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 61: 34–43, 2014     

A stochastic approximation algorithm to compute bid prices for joint capacity allocation and overbooking over an airline network

In this article, we develop a stochastic approximation algorithm to find good bid price policies for the joint capacity allocation and overbooking problem over an airline network. Our approach is based on visualizing the total expected profit as a function of the bid prices and searching for a good set of bid prices by using the stochastic gradients of the total expected profit function. We show that the total expected profit function that we use is differentiable with respect to the bid prices and derive a simple expression that can be used to compute its stochastic gradients. We show that the iterates of our stochastic approximation algorithm converge to a stationary point of the total expected profit function with probability 1. Our computational experiments indicate that the bid prices computed by our approach perform significantly better than those computed by standard benchmark strategies and the performance of our approach is relatively insensitive to the frequency with which we recompute the bid prices over the planning horizon. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Case‐based reasoning and improved adaptive search for project scheduling

Most scheduling problems are notoriously intractable, so the majority of algorithms for them are heuristic in nature. Priority rule‐based methods still constitute the most important class of these heuristics. Of these, in turn, parametrized biased random sampling methods have attracted particular interest, due to the fact that they outperform all other priority rule‐based methods known. Yet, even the “best” such algorithms are unable to relate to the full range of instances of a problem: Usually there will exist instances on which other algorithms do better. We maintain that asking for the one best algorithm for a problem may be asking too much. The recently proposed concept of control schemes, which refers to algorithmic schemes allowing to steer parametrized algorithms, opens up ways to refine existing algorithms in this regard and improve their effectiveness considerably. We extend this approach by integrating heuristics and case‐based reasoning (CBR), an approach that has been successfully used in artificial intelligence applications. Using the resource‐constrained project scheduling problem as a vehicle, we describe how to devise such a CBR system, systematically analyzing the effect of several criteria on algorithmic performance. Extensive computational results validate the efficacy of our approach and reveal a performance similar or close to state‐of‐the‐art heuristics. In addition, the analysis undertaken provides new insight into the behaviour of a wide class of scheduling heuristics. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 201–222, 2000     

The quay crane scheduling problem with time windows

The quay crane scheduling problem consists of scheduling tasks for loading and unloading containers on cranes that are assigned to a vessel for its service. This article introduces a new approach for quay crane scheduling, where the availability of cranes at a vessel is restricted to certain time windows. The problem is of practical relevance, because container terminal operators frequently redeploy cranes among vessels to speed up the service of high‐priority vessels while serving low‐priority vessels casually. This article provides a mathematical formulation of the problem and a tree‐search‐based heuristic solution method. A computational investigation on a large set of test instances is used to evaluate the performance of the heuristic and to identify the impact of differently structured crane time windows on the achievable vessel handling time. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011