Another greedy heuristic for the constrained forest problem

The constrained forest problem seeks a minimum-weight spanning forest in an undirected edge-weighted graph such that each tree spans at least a specified number of vertices. We present a greedy heuristic for this NP-hard problem, whose solutions are at least as good as, and often better than, those produced by the best-known 2-approximate heuristic.     

An effective heuristic for the CLSP with set-up times

The problem of multi-item, single level, capacitated, dynamic lot-sizing with set-up times (CLSP with set-up times) is considered. The difficulty of the problem compared with its counterpart without set-up times is explained. A lower bound on the value of the objective function is calculated by Lagrangian relaxation with subgradient optimisation. During the process, attempts are made to get good feasible solutions (ie. upper bounds) through a smoothing heuristic, followed by a local search with a variable neighbourhood. Solutions found in this way are further optimised by solving a capacitated transshipment problem. The paper describes the various elements of the solution procedure and presents the results of extensive numerical experimentation.     

An effective and fast heuristic for the Dial-a-Ride problem

Dial-a-Ride is an emerging transport system, in which a fleet of vehicles, without fixed routes and schedules, carries people from the desired pickup point to the desired delivery point, during a pre-specified time interval. It can be modeled as an -hard routing and scheduling problem, with a suitable mixed integer programming formulation. Exact approaches to this problem are too limited to tackle real-life instances (hundred of trips), therefore heuristics are needed. The heuristic method proposed in this paper builds an auxiliary graph and then solves an assignment problem on this graph. The auxiliary graph is obtained by replacing pairs of nodes with a single one and associating an ad hoc cost function to the new set of arcs. Two different simple methods are employed to transform the infeasible solution given by the assignment problem into a feasible one. The proposed algorithms have been tested on instances created using the Milan network and shown to be fast and effective.      

A fuzzy greedy heuristic for permutation flow-shop scheduling

This paper describes a polynomial-time heuristic for the permutation flow-shop scheduling problem with the makespan criterion. The proposed method consists of two phases: arranging the jobs in priority order and then constructing a sequence. A fuzzy greedy evaluation function is employed to prioritize the jobs for incorporating into the construction phase of the heuristic. Computational experiments using standard benchmark problems indicate an improvement of the new heuristic over the well-known Nawaz, Enscore and Ham (NEH) heuristic. It will be seen that the NEH heuristic is a special case of our more general heuristic.     

An application of the greedy heuristic of set cover to traffic checks

Let us suppose that X is a given, finite, not empty set and ${\mathcal F}Let us suppose that X is a given, finite, not empty set and is a given collection of subsets of X such that their union equals X, in other words covers X. Set cover is the problem of selecting as few as possible subsets from such that their union covers X. Max k-cover is the problem of selecting k subsets from such that their union has maximum cardinality. Both problems are NP-hard. There is a polynomial time greedy heuristic that iteratively selects the subset from that covers the largest number of yet uncovered elements. We implemented this greedy algorithm to support the planning of a checking system that is aimed to check the vehicles in a road network. We would like to answer such questions:

–	How many and which links are sufficient to check a given percentage of all traffic flow?
–	What percentage of traffic can be checked with given links?

This paper defines the necessary data and basic knowledge, gives algorithms to answer the previous questions and also shows the results of an implementation in a road network that contains about 11,000 junctions, 3,000 origin–destination junctions and 26,000 links.     

An effective heuristic for the two-dimensional irregular bin packing problem

This paper proposes an adaptation, to the two-dimensional irregular bin packing problem of the Djang and Finch heuristic (DJD), originally designed for the one-dimensional bin packing problem. In the two-dimensional case, not only is it the case that the piece’s size is important but its shape also has a significant influence. Therefore, DJD as a selection heuristic has to be paired with a placement heuristic to completely construct a solution to the underlying packing problem. A successful adaptation of the DJD requires a routine to reduce computational costs, which is also proposed and successfully tested in this paper. Results, on a wide variety of instance types with convex polygons, are found to be significantly better than those produced by more conventional selection heuristics.     

An effective and simple heuristic for the set covering problem

This paper investigates the development of an effective heuristic to solve the set covering problem (SCP) by applying the meta-heuristic Meta-RaPS (Meta-heuristic for Randomized Priority Search). In Meta-RaPS, a feasible solution is generated by introducing random factors into a construction method. Then the feasible solutions can be improved by an improvement heuristic. In addition to applying the basic Meta-RaPS, the heuristic developed herein integrates the elements of randomizing the selection of priority rules, penalizing the worst columns when the searching space is highly condensed, and defining the core problem to speedup the algorithm. This heuristic has been tested on 80 SCP instances from the OR-Library. The sizes of the problems are up to 1000 rows × 10,000 columns for non-unicost SCP, and 28,160 rows × 11,264 columns for the unicost SCP. This heuristic is only one of two known SCP heuristics to find all optimal/best known solutions for those non-unicost instances. In addition, this heuristic is the best for unicost problems among the heuristics in terms of solution quality. Furthermore, evolving from a simple greedy heuristic, it is simple and easy to code. This heuristic enriches the options of practitioners in the optimization area.     

A greedy heuristic for the capacitated minimum spanning tree problem

This paper develops a greedy heuristic for the capacitated minimum spanning tree problem (CMSTP), based on the two widely known methods of Prim and of Esau–Williams. The proposed algorithm intertwines two-stages: an enhanced combination of the Prim and Esau–Williams approaches via augmented and synthetic node selection criteria, and an increase of the feasible solution space by perturbing the input data using the law of cosines. Computational tests on benchmark problems show that the new heuristic provides extremely good performance results for the CMSTP, justifying its effectiveness and robustness. Furthermore, excluding the feasible space expansion, we show that we can still obtain good quality solutions in very short computational times.     

An adaptive large neighborhood search heuristic for the Pollution-Routing Problem

The Pollution-Routing Problem (PRP) is a recently introduced extension of the classical Vehicle Routing Problem with Time Windows which consists of routing a number of vehicles to serve a set of customers, and determining their speed on each route segment so as to minimize a function comprising fuel, emission and driver costs. This paper presents an adaptive large neighborhood search for the PRP. Results of extensive computational experimentation confirm the efficiency of the algorithm.     

An effective heuristic for large-scale capacitated facility location problems

The Capacitated Facility Location Problem (CFLP) consists of locating a set of facilities with capacity constraints to satisfy the demands of a set of clients at the minimum cost. In this paper we propose a simple and effective heuristic for large-scale instances of CFLP. The heuristic is based on a Lagrangean relaxation which is used to select a subset of “promising” variables forming the core problem and on a Branch-and-Cut algorithm that solves the core problem. Computational results on very large scale instances (up to 4 million variables) are reported.     

Sensitivity analysis of a greedy heuristic for knapsack problems

In this paper, we carry out parametric analysis as well as a tolerance limit based sensitivity analysis of a greedy heuristic for two knapsack problems—the 0–1 knapsack problem and the subset sum problem. We carry out the parametric analysis based on all problem parameters. In the tolerance limit based approach, we use a definition of the sensitivity analysis problem that is polynomial for polynomial heuristics. One of the interesting and counterintuitive results described in this paper is that the tolerance limits obtained from the heuristic sensitivity analysis cannot be used as bounds for the tolerance limits of the sensitivity analysis of optimal solutions in most cases.     

An effective quasi-human based heuristic for solving the rectangle packing problem

In this paper, we introduce an effective deterministic heuristic, Less Flexibility First, for solving the classical NP-complete rectangle packing problem. Many effective heuristics implemented for this problem are CPU-intensive and non-deterministic in nature. Others, including the polynomial approximation methodology [J. Assoc. Comput. Mach. 32 (1) (1985) 130] are too laborious for practical problem sizes. The technique we propose is inspired and developed by enhancing some rule-of-thumb guidelines resulting from the generation-long work experience of human professionals in ancient days. Although the Less Flexibility First heuristic is a deterministic algorithm, the results are very encouraging. This algorithm can consistently produce packing densities of around 99% on most randomly generated large examples. As compared with the recent results of a well known simulated annealing based Rectangle Packing (RP) algorithm [IEEE Trans. Computer-aided Design Integrated Circuits Systems 17 (1) (1998) 60], the results are much better both in less dead space² (4% vs 6.7%) and much less CPU time (9.57 vs 331.78 seconds). Experimenting our heuristics on a public rectangle packing data set covering instances of 16–97 rectangles, the average unpack ratio is quite satisfactory (0.92% for bounding boxes limited to be optimum and 2.68% for the completed packing), while most cases spend only a few minutes in CPU time.     

On parallelizing a greedy heuristic for finding small dominant sets

We analyse a greedy heuristic for finding small dominating sets in graphs: bounds on the size of the dominating set so produced had previously been derived in terms of the size of a smallest dominating set and the number of vertices and edges in the graph, respectively, We show that computing the resulting small dominating set isP-hard and so cannot be done efficiently in parallel (in the context of the PRAM model of parallel computation). We also consider a related non-deterministic greedy heuristic.     

An effective heuristic for minimising makespan on unrelated parallel machines

Scheduling independent tasks on unrelated machines is a relatively difficult and challenging problem. In this paper, we develop a tabu search based heuristic for minimising makespan for the above problem that can provide good quality solutions for practical size problems within a reasonable amount of computational time. Our adaptation of this tabu search uses hashing to control the tabu restrictions of the search process and requires fewer critical parameters than many of the common tabu search approaches employed for combinatorial optimisation. Hashing is simple to implement and very effective in providing a near-optimal solution. Computational results are presented to demonstrate the effectiveness of the proposed heuristic.     

A greedy look-ahead heuristic for the vehicle routing problem with time windows

In this paper we consider the problem of physically distributing finished goods from a central facility to geographically dispersed customers, which pose daily demands for items produced in the facility and act as sales points for consumers. The management of the facility is responsible for satisfying all demand, and promises deliveries to the customers within fixed time intervals that represent the earliest and latest times during the day that a delivery can take place. We formulate a comprehensive mathematical model to capture all aspects of the problem, and incorporate in the model all critical practical concerns such as vehicle capacity, delivery time intervals and all relevant costs. The model, which is a case of the vehicle routing problem with time windows, is solved using a new heuristic technique. Our solution method, which is based upon Atkinson's greedy look-ahead heuristic, enhances traditional vehicle routing approaches, and provides surprisingly good performance results with respect to a set of standard test problems from the literature. The approach is used to determine the vehicle fleet size and the daily route of each vehicle in an industrial example from the food industry. This actual problem, with approximately two thousand customers, is presented and solved by our heuristic, using an interface to a Geographical Information System to determine inter-customer and depot–customer distances. The results indicate that the method is well suited for determining the required number of vehicles and the delivery schedules on a daily basis, in real life applications.     

An Iterated Local Search heuristic for the Heterogeneous Fleet Vehicle Routing Problem

This paper deals with the Heterogeneous Fleet Vehicle Routing Problem (HFVRP). The HFVRP is $\mathcal{NP}$ -hard since it is a generalization of the classical Vehicle Routing Problem (VRP), in which clients are served by a heterogeneous fleet of vehicles with distinct capacities and costs. The objective is to design a set of routes in such a way that the sum of the costs is minimized. The proposed algorithm is based on the Iterated Local Search (ILS) metaheuristic which uses a Variable Neighborhood Descent procedure, with a random neighborhood ordering (RVND), in the local search phase. To the best of our knowledge, this is the first ILS approach for the HFVRP. The developed heuristic was tested on well-known benchmark instances involving 20, 50, 75 and 100 customers. These test-problems also include dependent and/or fixed costs according to the vehicle type. The results obtained are quite competitive when compared to other algorithms found in the literature.     

A robust heuristic for the Generalized Assignment Problem

The Generalized Assignment Problem, in the class of NP-hard problems, occurs in a wide range of applications — vehicle packing, computers, and logistics, to name only a few. Previous research has been concentrated on optimization methodologies for the GAP. Because the Generalized Assignment Problem is NP-hard, optimization methods tend to require larger computation times for large-scale problems. This paper presents a new heuristic,Variable-Depth-Search Heuristic (VDSH). We show that on the sets of large test problems, the quality of the solution found by VDSH exceeds that of the leading heuristic by an average of over twenty percent, while maintaining acceptable solution times. On difficult problem instances, VDSH provides solutions having costs 140% less than those found by the leading heuristic. A duality gap analysis of VDSH demonstrates the robustness of our heuristics.