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In automated laboratories consisting of multiple different types of instruments, scheduling algorithms are useful for determining the optimal allocations of instruments to minimize the time required to complete experimental procedures. However, previous studies on scheduling algorithms for laboratory automation have not emphasized the time constraints by mutual boundaries (TCMBs) among operations, which is important in procedures involving live cells or unstable biomolecules. Here, we define the "scheduling for laboratory automation in biology" (S-LAB) problem as a scheduling problem for automated laboratories in which operations with TCMBs are performed by multiple different instruments. We formulate an S-LAB problem as a mixed-integer programming (MIP) problem and propose a scheduling method using the branch-and-bound algorithm. Simulations show that our method can find the optimal schedules of S-LAB problems that minimize overall execution time while satisfying the TCMBs. Furthermore, we propose the use of our scheduling method for the simulation-based design of job definitions and laboratory configurations.

This research studies a production-recycling inventory model for postconsumer recycled content products made from a mix of virgin and recycled materials with specific proportions. The proportion is known as the postconsumer recycled content level. We study a closed-loop supply chain (CLSC) where a recycling facility and a supplier respectively ship the recycled and virgin raw materials to a manufacturer. The manufacturer mixes them based on the proportion to produce these products for a retailer. The recycling facility recycles the used products collected by a material recovery facility. The paper develops a joint economic lot-size model (JELS) to maximize the chain-wide total profit. A branch-and-bound algorithm is developed, along with the convex analysis to find the exact solution. To coordinate the CLSC, we present models called disproportionate profit-sharing (DPS) and proportional profit-sharing (PPS) operating under quantity discount. The DPS would give more powerful members, a higher proportion of the extra profits resulting from the JELS. However, the PPS guarantees equitable increases in the individual profits accruing to all members from the JELS. Whereas the DPS works properly in vertically integrated CLSCs and decentralized ones with dominant player(s), the PPS is appropriate for decentralized ones with no dominant players. Additionally, we prove the optimal level of postconsumer recycled content is often either at its maximum or minimum allowable. However, the zero-derivative points within the range can be optimum in a particular case. Consequently, we present decision trees for managers to determine the right profit-sharing method and the optimal postconsumer recycled content level.

Due to serious air pollution, some machines may be forced to shut down for short periods. During such periods, companies will make no progress on the yields. Maintenance also leads to reductions. If we view such two indispensable things such as shutdown and maintenance as jobs, the two can overlap with each other, i.e., the schedule can be shorter. However, this phenomenon is seldom discussed in scheduling. In this paper, we consider a three-agent scheduling problem with the overlap effect. The objective is to minimize the total tardiness of jobs from agent 3 under the restrictions that the PM 2.5 emissions of agent 1 cannot exceed a limit and that each maintenance activity requested by agent 2 must be conducted within its specific maintenance window. A branch-and-bound algorithm is developed to generate the optimal schedules. Computational experiments are conducted to show the solution quality and execution speed of the proposed algorithm.

Long queues of arrival trucks are a common problem in seaports, and thus, carbon emissions generated from trucks in the queue cause environmental pollution. In order to relieve gate congestion and reduce carbon emissions, this paper proposes a lane allocation framework combining the truck appointment system (TAS) for four types of trucks. Based on the distribution of arrival times obtained from the TAS, lane allocation decisions in each appointment period are determined in order to minimize the total cost, including the operation cost and carbon emissions cost. The resultant optimization model is a non-linear fractional integer program. This model was firstly transformed to an equivalent integer program with bilinear constraints. Then, an improved branch-and-bound algorithm was designed, which includes further transforming the program into a linear program using the McCormick approximation method and iteratively generating a tighter outer approximation along the branch-and-bound procedure. Numerical studies confirmed the validity of the proposed model and algorithm, while demonstrating that the lane allocation decisions could significantly reduce carbon emissions and operation costs.

In this study, we consider an airport gate assignment problem that assigns a set of aircraft to a set of gates. The aircraft that cannot be assigned to any gate are directed to an apron. We aim to make aircraft-gate assignments so as to minimize the number of aircraft assigned to apron and among the apron usage minimizing solutions, we aim to minimize total walking distance travelled by all passengers. The problem is formulated as a mixed-integer nonlinear programming model and then it is linearized. A branch and bound algorithm, beam search and filtered beam search algorithms that employ powerful lower and upper bounding mechanisms are developed. The results of the computational experiment have shown the satisfactory performance of the algorithms.

Current generalizations of the central ideas of single-objective branch-and-bound to the multiobjective setting do not seem to follow their train of thought all the way. The present paper complements the various suggestions for generalizations of partial lower bounds and of overall upper bounds by general constructions for overall lower bounds from partial lower bounds, and by the corresponding termination criteria and node selection steps. In particular, our branch-and-bound concept employs a new enclosure of the set of nondominated points by a union of boxes. On this occasion we also suggest a new discarding test based on a linearization technique. We provide a convergence proof for our general branch-and-bound framework and illustrate the results with numerical examples.

We study the Knapsack Problem with Conflicts, a generalization of the Knapsack Problem in which a set of conflicts specifies pairs of items which cannot be simultaneously selected. In this work, we propose a novel combinatorial branch-and-bound algorithm for this problem based on an n-ary branching scheme. Our algorithm effectively combines different procedures for pruning the branch-and-bound nodes based on different relaxations of the Knapsack Problem with Conflicts. Its main elements of novelty are: (i) the adoption of the branching-and-pruned set branching scheme which, while extensively used in the maximum-clique literature, was never successfully employed for solving the Knapsack Problem with Conflicts; (ii) the adoption of the Multiple-Choice Knapsack Problem for the derivation of upper bounds used for pruning the branch-and-bound tree nodes; and (iii) the design of a new upper bound for the latter problem which can be computed very efficiently. Key to our algorithm is its high pruning potential and the low computational effort that it requires to process each branch-and-bound node. An extensive set of experiments carried out on the benchmark instances typically used in the literature shows that, for edge densities ranging from 0.1 to 0.9, our algorithm is faster by up to two orders of magnitude than the state-of-the-art method and by up to several orders of magnitude than a state-of-the-art mixed-integer linear programming solver., This work has been partially funded by the Spanish Ministry of Science, Innovation and Universities through the project COGDRIVE (DPI2017-86915-C3-3-R).

We consider the problem of scheduling a set of jobs subject to unequal release dates on a two-machine flow shop where the no-wait and non-availability constraints are considered so as to minimize the makespan. The contribution of this paper is two-fold. First, we propose a new mathematical formulation for the problem and derive valid inequalities. Second, we propose new lower bounds that are based on single and two-machine relaxations. These lower bounds are embedded onto a branch-and-bound algorithm enhanced by elimination and dominance rules. Computational results show that our exact methods consistently outperform the state-of-the-art branch-and-bound procedure.

The ordered weighted averaging (OWA) operator is one of the most used techniques in the operator’s aggregation procedure. This paper proposes a new assignment algorithm by using the OWA operator and different extensions of it in the Branch-and-bound algorithm. The process is based on the use of the ordered weighted average distance operator (OWAD) and the induced OWAD operator (IOWAD). We present it as the Branch-and-bound algorithm with the OWAD operator (BBAOWAD) and the Branch-and-bound algorithm with the IOWAD operator (BBAIOWAD). The main advantage of this approach is that we can obtain more detailed information by obtaining a parameterized family of aggregation operators. The application of the new algorithm is developed in a consumer decision-making model in the city of Barcelona regarding the selection of groceries by districts that best suit their needs. We rely on the opinion of local commerce experts in the city. The key advantage of this approach is that we can consider different sources of information independent of each other.

We consider a generalization of the knapsack problem in which items are partitioned into classes, each characterized by a fixed cost and capacity. We study three alternative Integer Linear Programming formulations. For each formulation, we design an efficient algorithm to compute the linear programming relaxation (one of which is based on Column Generation techniques). We theoretically compare the strength of the relaxations and derive specific results for a relevant case arising in benchmark instances from the literature. Finally, we embed the algorithms above into a unified implicit enumeration scheme which is run in parallel with an improved Dynamic Programming algorithm to effectively solve the problem to proven optimality. An extensive computational analysis shows that our new exact algorithm is capable of efficiently solving all the instances of the literature and turns out to be the best algorithm for instances with a low number of classes.

In the context of recreational routing, the problem of finding a route which starts and ends in the same location (while achieving a length between specified upper and lower boundaries) is a common task, especially for tourists or cyclists who want to exercise. The topic of finding a tour between a specified starting and ending location while minimizing one or multiple criteria is well covered in literature. In contrast to this, the route planning task in which a pleasant tour with length between a maximum and a minimum boundary needs to be found is relatively underexplored. In this paper, we provide a formal definition of this problem, taking into account the existing literature on which route attributes influence cyclists in their route choice. We show that the resulting problem is NP-hard and devise a branch-and-bound algorithm that is able to provide a bound on the quality of the best solution in pseudo-polynomial time. Furthermore, we also create an efficient heuristic to tackle the problem and we compare the quality of the solutions that are generated by the heuristic with the bounds provided by the branch-and-bound algorithm. Also, we thoroughly discuss the complexity and running time of the heuristic.

International audience; This paper deals with the cyclic job shop problem where the task durations are uncertain and belong to a polyhedral uncertainty set. We formulate the cyclic job shop problem as a two-stage robust optimization model. The cycle time and the execution order of tasks executed on the same machines correspond to the here-and-now decisions and have to be decided before the realization of the uncertainty. The starting times of tasks corresponding to the wait-and-see decisions are delayed and can be adjusted after the uncertain parameters are known. In the last decades, different solution approaches have been developed for two-stage robust optimization problems. Among them, the use of affine policies, column generation algorithms, row and row-and-column generation algorithms. In this paper, we propose a Branch-and-Bound algorithm to tackle the robust cyclic job shop problem with cycle time minimization. The algorithm uses, at each node of the search tree, a robust version of the Howard’s algorithm to derive a lower bound on the optimal cycle time. We also develop a heuristic method that permits to compute an initial upper bound for the cycle time. Finally, encouraging preliminary results on numerical experiments performed on randomly generated instances are presented.

International audience; We propose a solution approach for the general problem (QP) of minimizing a quadratic function of bounded integer variables subject to a set of quadratic constraints. The resolution is based on the reformulation of the original problem (QP) into an equivalent quadratic problem whose continuous relaxation is convex, so that it can be effectively solved by a branch-and-bound algorithm based on quadratic convex relaxation. We concentrate our efforts on finding a reformulation such that the continuous relaxation bound of the reformulated problem is as tight as possible. Furthermore, we extend our method to the case of mixed-integer quadratic problems with the following restriction: all quadratic sub-functions of purely continuous variables are already convex. Finally, we illustrate the different results of the article by small examples and we present some computational experiments on pure-integer and mixed-integer instances of (QP). Most of the considered instances with up to 53 variables can be solved by our approach combined with the use of Cplex.

A new discrete network design problem (DNDP) was pro-posed in this paper, where the variables can be a series of integers rather than just 0-1. The new DNDP can determine both capacity improvement grades of reconstruction roads and locations and capacity grades of newly added roads, and thus complies with the practical projects where road capacity can only be some discrete levels corresponding to the number of lanes of roads. This paper designed a solution algorithm combining branch-and-bound with Hooke-Jeeves algorithm, where feasible integer solutions are recorded in searching the process of Hooke-Jeeves algorithm, lend -ing itself to determine the upper bound of the upper-level problem. The thresholds for branch cutting and ending were set for earlier convergence. Numerical examples are given to demonstrate the efficiency of the proposed algorithm.

Preference rankings virtually appear in all fields of science (political sciences, behavioral sciences, machine learning, decision making and so on). The well-known social choice problem consists in trying to find a reasonable procedure to use the aggregate preferences or rankings expressed by subjects to reach a collective decision. This turns out to be equivalent to estimate the consensus (central) ranking from data and it is known to be a NP-hard problem. A useful solution has been proposed by Emond and Mason in 2002 through the Branch-and-Bound algorithm (BB) within the Kemeny and Snell axiomatic framework. As a matter of fact, BB is a time demanding procedure when the complexity of the problem becomes untractable, i.e. a large number of objects, with weak and partial rankings, in presence of a low degree of consensus. As an alternative, we propose an accurate heuristic algorithm called FAST that finds at least one of the consensus ranking solutions found by BB saving a lot of computational time. In addition, we show that the building block of FAST is an algorithm called QUICK that finds already one of the BB solutions so that it can be fruitfully considered to speed up even more the overall searching procedure if the number of objects is low. Simulation studies and applications on real data allows to show the accuracy and the computational efficiency of our proposal.

International audience; In this paper, we consider the two-machine no-wait flow-shop scheduling problem, when every machine is subject to one non-availability constraint and jobs have different release dates. The non-availability intervals of the machines overlap and they are known in advance. We aim to find a non-resumable schedule that minimizes the makespan. We propose several lower bounds and upper bounds. These bounding procedures are used in a branch-and-bound algorithm. Computational experiments are carried out on a large set of instances and the obtained results show the effectiveness of our method.

The purpose of this paper is to propose a new branch-and-bound algorithm for a class of scheduling problems to minimize total tardiness on identical parallel machines. In this algorithm, the Lagrangian relaxation technique is applied to obtain a tight lower bound. In addition, the job dominance conditions for the single machine total tardiness problem are utilized for both restricting branches and improving the lower bound. As will be shown by computational experiments, instances with up to 25 jobs and with any number of machines are optimally solved by the proposed algorithm.

In this study we consider the elevator operation problem of single-car elevator systems with destination hall call registration. In this part we construct a branch-and-bound algorithm to solve the dynamic operation optimization problem formulated in the first part. To calculate lower bounds of the subproblems generated in the course of the branch-and-bound algorithm, we first relax some of the constraints of the subproblems and decompose the relaxed subproblems into three parts. Then, we apply the Lagrangian relaxation method to the decomposed subproblems.

The resource-constrained project-scheduling problem (RCPSP) and some of its generalizations are defined. Furthermore, constraint propagation techniques for these problems and related machine scheduling problems are introduced and possible applications of these techniques in connection with lower bound calculations, branch-and-bound methods, and heuristics are discussed.

By considering two criteria of minimum total sum of vagueness and minimum total sum of squares in estimation, this article proposes a variable selection method for a fuzzy regression equation with crisp-input and fuzzy-output. A branch-and-bound algorithm is designed and “the least resistance principle” is adopted to determine the set of compromised solutions. Numerical examples are provided for illustration.

We solve for the first time to proven optimality the small instances in the classical literature benchmark of Minimum Linear Arrangement. This is achieved by formulating the problem as an ILP in a somehow unintuitive way, using variables expressing the fact that a vertex is between two other adjacent vertices in the arrangement. Using (only) these variables appears to be the key idea of the approach. Indeed, with these variables already the use of very simple constraints leads to good results, which can however be improved with a more detailed study of the underlying polytope.

Many problems in economy may be formulated as global optimization problems. Most numerically promising methods for solution of multivariate unconstrained Lipschitz optimization problems of dimension greater than 2 use rectangular or simplicial branch‐and‐bound techniques with computationally cheap, but rather crude lower bounds. The proposed branch‐and‐bound algorithm with simplicial partitions for global optimization uses a combination of 2 types of Lipschitz bounds. One is an improved Lipschitz bound with the first norm. The other is a combination of simple bounds with different norms. The efficiency of the proposed global optimization algorithm is evaluated experimentally and compared with the results of other well‐known algorithms. The proposed algorithm often outperforms the comparable branch‐and‐bound algorithms. Globalusis optimizavimas šakų ir rėžių algoritmu su lipšico rėžių junginiu simplekse Santrauka.Daug įvairių ekonomikos uždavinių yra formuluojami kaip globaliojo optimizavimo uždaviniai. Didžiojidalis Lipšico globaliojo optimizavimo metodų, tinkamų spręsti didesnės dimensijos, t. y. n > 2, uždavinius,naudoja stačiakampį arba simpleksinį šakų ir rėžių metodus bei paprastesnius rėžius. Šiame darbepasiūlytas simpleksinis šakų ir rėžių algoritmas, naudojantis dviejų tipų viršutinių rėžių junginį. Pirmasisyra pagerintas rėžis su pirmąja norma, kitas – trijų paprastesnių rėžių su skirtingomis normomis junginys.Gautieji eksperimentiniai pasiūlyto algoritmo rezultatai yra palyginti su kitų gerai žinomų Lipšico optimizavimo algoritmų rezultatais. Reikšminiai žodžiai: šakų ir rėžių algoritmas, globalusis optimizavimas, Lipšico optimizavimas, Lipšicorėžis. First published online:21 Oct 2010

International audience; This article investigates identical parallel machines scheduling with family setup times. The objective function being the weighted sum of completion times, the problem is known to be strongly NP-hard. We propose a constructive heuristic algorithm and three complementary lower bounds. Two of these bounds proceed by elimination of setup times or by distributing each of them to jobs of the corresponding family, while the third one is based on a lagrangian relaxation. The bounds and the heuristic are incorporated into a branch-and-bound algorithm. Experimental results obtained outperform those of the methods presented in previous works, in term of size of solved problems.

We address the NP-hard single-machine problem of scheduling n independent jobs so as to minimize the sum of α times total completion time and β times total earliness with β > α, which can be rewritten as an earliness–tardiness problem. Postponing jobs by leaving the machine idle may then be advantageous. The allowance of machine idle time between the execution of jobs singles out our problem from most concurrent research on problems with earliness penalties. Solving the problem to optimality poses a computational challenge, since the possibility of leaving the machine idle has a major effect on designing a branch-and-bound algorithm in general, and on computing lower bounds in particular. We present a branch-and-bound algorithm which is based upon many dominance rules and various lower bound approaches, including relaxation of the machine capacity, data manipulation, and Lagrangian relaxation. The algorithm is shown to solve small instances with up to 20 jobs.

This paper investigates algorithm development and implementation for multicriteria and multiconstraint level (MC2) integer linear programming problems. MC2 linear programming is an extension of linear programming (LP) and multiple criteria (MC) linear programming and a promising computer-aided decision technique in many applications. Here, we present two of the most recent techniques, the MC2 branch-and-partition algorithm and the MC2 branch-and-bound algorithm, to solve MC2 integer linear programs. We describe the design and implementation of a C++ software library for these approaches, and then conduct a comparison study in terms of computational efficiency and complexity through a series of empirical tests.