A Lagrangian relax-and-cut approach for the sequential ordering problem with precedence relationships

The sequential ordering problem with precedence relationships was introduced in Escudero [7]. It has a broad range of applications, mainly in production planning for manufacturing systems. The problem consists of finding a minimum weight Hamiltonian path on a directed graph with weights on the arcs, subject to precedence relationships among nodes. Nodes represent jobs (to be processed on a single machine), arcs represent sequencing of the jobs, and the weights are sums of processing and setup times. We introduce a formulation for the constrained minimum weight Hamiltonian path problem. We also define Lagrangian relaxation for obtaining strong lower bounds on the makespan, and valid cuts for further tightening of the lower bounds. Computational experience is given for real-life cases already reported in the literature.     

An exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts

This paper presents a new exact algorithm for the Capacitated Vehicle Routing Problem (CVRP) based on the set partitioning formulation with additional cuts that correspond to capacity and clique inequalities. The exact algorithm uses a bounding procedure that finds a near optimal dual solution of the LP-relaxation of the resulting mathematical formulation by combining three dual ascent heuristics. The first dual heuristic is based on the q-route relaxation of the set partitioning formulation of the CVRP. The second one combines Lagrangean relaxation, pricing and cut generation. The third attempts to close the duality gap left by the first two procedures using a classical pricing and cut generation technique. The final dual solution is used to generate a reduced problem containing only the routes whose reduced costs are smaller than the gap between an upper bound and the lower bound achieved. The resulting problem is solved by an integer programming solver. Computational results over the main instances from the literature show the effectiveness of the proposed algorithm.      

An linear programming based lower bound for the simple assembly line balancing problem

The simple assembly line balancing problem is a classical integer programming problem in operations research. A set of tasks, each one being an indivisible amount of work requiring a number of time units, must be assigned to workstations without exceeding the cycle time. We present a new lower bound, namely the LP relaxation of an integer programming formulation based on Dantzig–Wolfe decomposition. We propose a column generation algorithm to solve the formulation. Therefore, we develop a branch-and-bound algorithm to exactly solve the pricing problem. We assess the quality of the lower bound by comparing it with other lower bounds and the best-known solution of the various instances from the literature. Computational results show that the lower bound is equal to the best-known objective function value for the majority of the instances. Moreover, the new LP based lower bound is able to prove optimality for an open problem.     

Minimizing the completion time of a project under resource constraints and feeding precedence relations: a Lagrangian relaxation based lower bound

In this paper we study the Resource Constrained Project Scheduling Problem (RCPSP) with “Feeding Precedence” (FP) constraints and minimum makespan objective. This problem typically arises in production planning environment, like make-to-order manufacturing, where the effort associated with the execution of an activity is not univocally related to its duration percentage and the traditional finish-to-start precedence constraints or the generalized precedence relations cannot completely represent the overlapping among activities. In this context, we need to introduce in the RCPSP the FP constraints. For this problem we propose a new mathematical formulation and define a lower bound based on the Lagrangian relaxation of the resource constraints. A computational experimentation on randomly generated instances of sizes of up to 100 activities shows a better performance of this lower bound when compared to other lower bounds. Moreover, for the optimally solved instances, its value is very close to the optimal one. Furthermore, in order to show the effectiveness of the proposed lower bound on large instances for which the optimal solution is known, we adapted our approach to solve the benchmarks of the basic RCPSP from the PSLIB with 120 activities.     

Lower bounds for the earliness–tardiness scheduling problem on parallel machines with distinct due dates

This paper addresses the parallel machine scheduling problem in which the jobs have distinct due dates with earliness and tardiness costs. New lower bounds are proposed for the problem, they can be classed into two families. First, two assignment-based lower bounds for the one-machine problem are generalized for the parallel machine case. Second, a time-indexed formulation of the problem is investigated in order to derive efficient lower bounds throught column generation or Lagrangean relaxation. A simple local search algorithm is also presented in order to derive an upper bound. Computational experiments compare these bounds for both the one machine and parallel machine problems and show that the gap between upper and lower bounds is about 1.5%.     

An exact algorithm to minimize the makespan in project scheduling with scarce resources and generalized precedence relations

In this paper we propose an exact algorithm for the Resource Constrained Project Scheduling Problem (RCPSP) with generalized precedence relationships (GPRs) and minimum makespan objective. For the RCPSP with GPRs we give a new mathematical formulation and a branch and bound algorithm exploiting such a formulation. The exact algorithm takes advantage also of a lower bound based on a Lagrangian relaxation of the same mathematical formulation. We provide an extensive experimentation and a comparison with known lower bounds and competing exact algorithms drawn from the state of the art.     

On due-date based valid cuts for the sequential ordering problem

The Sequential Ordering Problem with precedence relationships was introduced in Escudero (1988), and extended to cover release and due dates in Escudero and Schiomachen (1993). The problem consists of finding a minimum weight Hamiltonian path on a directed graph with weights on the nodes and the arcs, satisfying precedence relationships among the nodes and lower and upper bounds on the Hamiltonian subpaths. In this paper we introduce valid cuts derived from the due date constraints that can be used in a separation framework for the dual (Lagrangian based) relaxation of the problem. We also provide an heuristic separation algorithm to obtain those cuts. This research was supported by DGICYT through grant N. PB95-0407     

The equal flow problem

This paper presents a new algorithm for the solution of a network problem with equal flow side constraints. The solution technique is motivated by the desire to exploit the special structure of the side constraints and to maintain as much of the characteristics of pure network problems as possible. The proposed algorithm makes use of Lagrangean relaxation to obtain a lower bound and decomposition by right-hand-side allocation to obtain upper bounds. The lagrangean dual serves not only to provide a lower bound used to assist in termination criteria for the upper bound, but also allows an initial allocation of equal flows for the upper bound. The algorithm has been tested on problems with up to 1500 nodes and 6000 arcs. Computational experience indicates that solutions whose objective function value is well within 1% of the optimum can be obtained in 1%–65% of the MPSX time depending on the amount of imbalance inherent in the problem. Incumbent integer solutions which are within 99.99% feasible and well within 1% of the proven lower bound are obtained in a straightforward manner requiring, on the average, 30% of the MPSX time required to obtain a linear optimum.     

A new exact method for the two-dimensional orthogonal packing problem

The two-dimensional orthogonal packing problem (2OPP) consists in determining if a set of rectangles (items) can be packed into one rectangle of fixed size (bin). In this paper we propose two exact algorithms for solving this problem. The first algorithm is an improvement on a classical branch&bound method, whereas the second algorithm is based on a new relaxation of the problem. We also describe reduction procedures and lower bounds which can be used within enumerative methods. We report computational experiments for randomly generated benchmarks which demonstrate the efficiency of both methods: the second method is competitive compared to the best previous methods. It can be seen that our new relaxation allows an efficient detection of non-feasible instances.     

The flow circulation sharing problem

The flow circulation sharing problem is defined as a network flow circulation problem with a maximin objective function. The arcs in the network are partitioned into regular arcs and tradeoff arcs where each tradeoff arc has a non-decreasing tradeoff function associated with it. All arcs have lower and upper bounds on their flow while the value of the smallest tradeoff function is maximized. The model is useful in equitable resource allocation problems over time which is illustrated in a coal strike example and a submarine assignment example. Some properties including optimality conditions are developed. Each cut in the network defines a knapsack sharing problem which leads to an optimality condition similar to the max flow/min cut theorem. An efficient algorithm for both the continuous and integer versions of the flow circulation sharing problem is developed and computational experience given. In addition, efficient algorithms are developed for problems where some of the arcs have infinite flow upper bounds.     

Local networks for computer communications: A. WEST and P. JANSON (Eds.) Proceedings of the IFIP Working Group 6.4, International Workshop on Local Networks,Zürich,Switzerland, August 27–29, 1980, North-Holland,Amsterdam, 1981, xiii + 470 pages,Dfl.125.00

In this paper we present two lower bounds for the p-median problem, the problem of locating p facilities (medians) on a network. These bounds are based on two separate lagrangean relaxations of a zero-one formulation of the problem with subgradient optimisation being used to maximise these bounds. Penalty tests based on these lower bounds and a heuristically determined upper bound to the problem are developed and shown to result in a large reduction in problem size. The incorporation of the lower bounds and the penalty tests into a tree search procedure is described and computational results are given for problems with an arbitrary number of medians and having up to 200 vertices. A comparison is also made between these algorithms and the dual-based algorithm of Erlenkotter.     

Optimal toll design: a lower bound framework for the asymmetric traveling salesman problem

We propose a framework of lower bounds for the asymmetric traveling salesman problem (TSP) based on approximating the dynamic programming formulation with different basis vector sets. We discuss how several well-known TSP lower bounds correspond to intuitive basis vector choices and give an economic interpretation wherein the salesman must pay tolls as he travels between cities. We then introduce an exact reformulation that generates a family of successively tighter lower bounds, all solvable in polynomial time. We show that the base member of this family yields a bound greater than or equal to the well-known Held-Karp bound, obtained by solving the linear programming relaxation of the TSP’s integer programming arc-based formulation.     

An exact algorithm for a vehicle routing problem with time windows and multiple use of vehicles

The vehicle routing problem with multiple use of vehicles is a variant of the classical vehicle routing problem. It arises when each vehicle performs several routes during the workday due to strict time limits on route duration (e.g., when perishable goods are transported). The routes are defined over customers with a revenue, a demand and a time window. Given a fixed-size fleet of vehicles, it might not be possible to serve all customers. Thus, the customers must be chosen based on their associated revenue minus the traveling cost to reach them. We introduce a branch-and-price approach to address this problem where lower bounds are computed by solving the linear programming relaxation of a set packing formulation, using column generation. The pricing subproblems are elementary shortest path problems with resource constraints. Computational results are reported on euclidean problems derived from well-known benchmark instances for the vehicle routing problem with time windows.     

Lower and upper bounding procedures for the bin packing problem with concave loading cost

We address the one-dimensional bin packing problem with concave loading cost (BPPC), which commonly arises in less-than-truckload shipping services. Our contribution is twofold. First, we propose three lower bounds for this problem. The first one is the optimal solution of the continuous relaxation of the problem for which a closed form is proposed. The second one allows the splitting of items but not the fractioning of bins. The third one is based on a large-scale set partitioning formulation of the problem. In order to circumvent the challenges posed by the non-linearity of the objective function coefficients, we considered the inner-approximation of the concave load cost and derived a relaxed formulation that is solved by column generation. In addition, we propose two subset-sum-based heuristics. The first one is a constructive heuristic while the second one is a local search heuristic that iteratively attempts to improve the current solution by selecting pairs of bins and solving the corresponding subset sum-problem. We show that the worst-case performance of any BPPC heuristic and any concave loading cost function is bounded by 2. We present the results of an extensive computational study that was carried out on large set of benchmark instances. This study provides empirical evidence that the column generation-based lower bound and the local search heuristic consistently exhibit remarkable performance.     

Dual Bounds and Optimality Cuts for All-Quadratic Programs with Convex Constraints

A central problem of branch-and-bound methods for global optimization is that often a lower bound do not match with the optimal value of the corresponding subproblem even if the diameter of the partition set shrinks to zero. This can lead to a large number of subdivisions preventing the method from terminating in reasonable time. For the all-quadratic optimization problem with convex constraints we present optimality cuts which cut off a given local minimizer from the feasible set. We propose a branch-and-bound algorithm using optimality cuts which is finite if all global minimizers fulfill a certain second order optimality condition. The optimality cuts are based on the formulation of a dual problem where additional redundant constraints are added. This technique is also used for constructing tight lower bounds. Moreover we present for the box-constrained and the standard quadratic programming problem dual bounds which have under certain conditions a zero duality gap.     

A distributed exact algorithm for the multiple resource constrained sequencing problem

Sequencing problems arise in the context of process scheduling both in isolation and as subproblems for general scenarios. Such sequencing problems can often be posed as an extension of the Traveling Salesman Problem. We present an exact parallel branch and bound algorithm for solving the Multiple Resource Constrained Traveling Salesman Problem (MRCTSP), which provides a platform for addressing a variety of process sequencing problems. The algorithm is based on a linear programming relaxation that incorporates two families of inequalities via cutting plane techniques. Computational results show that the lower bounds provided by this method are strong for the types of problem generators that we considered as well as for some industrially derived sequencing instances. The branch and bound algorithm is parallelized using the processor workshop model on a network of workstations connected via Ethernet. Results are presented for instances with up to 75 cities, 3 resource constraints, and 8 workstations.     

Job-shop scheduling with resource-time models of operations

The job-shop problems with allocation of continuously-divisible nonrenewable resource is considered. The mathematical models of operations are linear, decreasing functions with respect to an amount of resource. The objective is sequencing operations and allocation of constrained resource such that the project duration is minimized. Thus, the problem considered is a generalization of the classical job-shop problem. Some properties of the optimal solution are presented. The algorithm of solving this problem is based on the disjunctive graphs theory and branch-and-bound technique. The theory of the algorithm is based on the critical path concept using the segment system approach. The special feature of the algorithm is that it gives a complete solution which is associated with each node of the enumeration tree. Possible generalizations of the results presented are indicated.     

求解非凸区域上凸函数比式和问题的全局优化方法

针对非凸区域上的凸函数比式和问题,给出一种求其全局最优解的确定性方法.该方法基于分支定界框架.首先通过引入变量,将原问题等价转化为d.c.规划问题,然后利用次梯度和凸包络构造松弛线性规划问题,从而将关键的估计下界问题转化为一系列线性规划问题,这些线性规划易于求解而且规模不变,更容易编程实现和应用到实际中;分支采用单纯形对分不但保证其穷举性,而且使得线性规划规模更小.理论分析和数值实验表明所提出的算法可行有效.     

一种基于匈牙利算法的二次分配问题求解方法

二次分配问题(Quadratic assignment problem,QAP)属于NP-hard组合优化难题.二次分配问题的线性化及下界计算方法,是求解二次分配问题的重要途径.以Frieze-Yadegar线性化模型和Gilmore-Lawler下界为基础,详细论述了二次分配问题线性化模型的结构特征,并分析了Gilmore-Lawler下界值往往远离目标函数最优值的原因.在此基础上,提出一种基于匈牙利算法的二次分配问题对偶上升下界求解法.通过求解QAPLIB中的部分实例,说明了方法的有效和可行性.     

A branch and bound method for the job-shop problem with sequence-dependent setup times

This paper deals with the job-shop scheduling problem with sequence-dependent setup times. We propose a new method to solve the makespan minimization problem to optimality. The method is based on iterative solving via branch and bound decisional versions of the problem. At each node of the branch and bound tree, constraint propagation algorithms adapted to setup times are performed for domain filtering and feasibility check. Relaxations based on the traveling salesman problem with time windows are also solved to perform additional pruning. The traveling salesman problem is formulated as an elementary shortest path problem with resource constraints and solved through dynamic programming. This method allows to close previously unsolved benchmark instances of the literature and also provides new lower and upper bounds.