Optimization and heuristic methods supporting distributed manufacturing

New optimization and heuristic methods are described to address supply chain management problems in distributed manufacturing settings. Specifically, integer programming formulations and heuristic methods are developed to design and evaluate optimal or near-optimal delivery plans for material movements between sites in a truckload trucking environment for the benefit of carriers, customers and professional drivers. The tools developed herein are appropriate for examining delivery needs between suppliers, manufacturers, distribution centres, and customer locations. They are equally applicable to more complex situations involving the return of packaging materials to the original shipment site, or even concurrent consideration of multiple business entities with various shipment profiles. Realistically sized case studies are provided to demonstrate the efficacy of the approaches using data supplied by J.B. Hunt Transport, Inc.     

Approaches to solve the fleet-assignment,aircraft-routing,crew-pairing and crew-rostering problems of a regional carrier

This paper presents the results of a research project funded by a regional carrier operating inter-island services within the Canary Islands (Spain) in addition to services to Morocco and Portugal. It operates between 100 and 150 flights a day using three airline operators. The main scope of the project was to solve fleet-assignment, aircraft-routing, crew-pairing and crew-rostering problems on real-world data. The special characteristics of the carrier, flying between 7 am and 11 pm every day, have motivated us to design models and algorithms that are different than the ones addressed in the literature, typically built for large airline companies. This paper shows a solution approach for an integrated fleet-assignment, aircraft-routing and crew-pairing problem covering the flights of a single day. This is a new combinatorial problem that can be considered as a 2-depot vehicle routing problem with driver changes, where the vehicles represent aircrafts and the drivers represent crews. Adapting approaches from the vehicle routing literature, this paper describes a heuristic algorithm based on an integer programming model. In a similar way, this paper also addresses the rostering problem. This problem can be decomposed in smaller problems taking into account operators, bases and crew groups. These problems admit a compact formulation through mixed integer linear programming models which can be tracked by modern general-purpose solvers. This paper illustrates the success of our solution approaches on real-world instances. The airline carrier is currently using these approaches.     

HEURISTICS FOR INTEGER PROGRAMMING USING SURROGATE CONSTRAINTS

This paper proposes a class of surrogate constraint heuristics for obtaining approximate, near optimal solutions to integer programming problems. These heuristics are based on a simple framework that illuminates the character of several earlier heuristic proposals and provides a variety of new alternatives. The paper also proposes additional heuristics that can be used either to supplement the surrogate constraint procedures or to provide independent solution strategies. Preliminary computational results are reported for applying one of these alternatives to a class of nonlinear generalized set covering problems involving approximately 100 constraints and 300–500 integer variables. The solutions obtained by the tested procedure had objective function values twice as good as values obtained by standard approaches (e.g., reducing the best objective function values of other methods from 85 to 40 on the average. Total solution time for the tested procedure ranged from ten to twenty seconds on the CDC 6600.     

Optimum turn-restricted paths,nested compatibility,and optimum convex polygons

We consider two apparently unrelated classes of combinatorial and geometric optimization problems. First, we give compact extended formulations, i.e., polynomial-size linear programming formulations with integer optima, for optimum path problems with turn restrictions satisfying a nested compatibility condition in acyclic digraphs. We then apply these results to optimum convex polygon problems in the plane, by interpreting certain dynamic programming algorithms as sequences of optimum turn-restricted path problems with nested compatibility in acyclic digraphs. As a result, we derive compact extended formulations for these geometric problems as well.     

INTEGER GOAL PROGRAMMING IN ADVERTISING MEDIA SELECTION

The media selection decision allocates advertising dollars among competing media so as to optimize promotional and corporate objectives. Linear programming attempts to model this process have been complicated by multiple and often conflicting management goals, the need for integer solutions, and nonlinearities. This study offers a technique that is sufficiently robust to simultaneously handle these problems. An alternative media selection framework is presented and the results of an illustrative application of integer goal programming are discussed. The proposed model improves on linear programming by success fully providing for optimal, integer solutions in settings that more realistically reflect the complexity of the media decision environment.     

Theory of integer-valued data envelopment analysis under alternative returns to scale axioms

Kuosmanen and Kazemi Matin [Theory of integer valued data envelopment analysis. European Journal of Operational Research 2009;192:658–67] developed an axiomatic foundation for a data envelopment analysis (DEA) model that assumes subsets of input and output variables to be integer valued. In this paper we extend and generalize the axiomatic foundation for the integer DEA under variable, non-decreasing and non-increasing returns to scale. These model variants are achieved by introducing new axioms of natural convexity and natural augmentability. We also develop mixed integer linear programming (MILP) formulations for computing efficiency scores in these environments. An empirical example illustrates the approach.     

Exact methods in optimum disassembly sequence search for problems subject to sequence dependent costs

Disassembling complex products is formally approached via network representation and subsequent mathematical modeling, aimed at selecting a good or optimum sequence of disassembly operations. This is done via heuristics, metaheuristics or mathematical programming. In contrast with heuristics and metaheuristics, which select a near-optimum solution, mathematical programming guarantees the selection of the global optimum. This problem is relatively simple if the disassembly costs can be assumed sequence independent. In practice, however, sequence dependent disassembly costs are frequently encountered, which causes NP-completeness of the problem. Although methods, e.g., based on the two-commodity network flow approach, are available to solve this constrained asymmetric Traveling Salesperson problem rigorously, this requires the introduction of integer variables. In this paper, a modification of the two-commodity network flow approach is proposed, which reduces the number of integer variables. This is applied to product structures that can be represented by a disassembly precedence graph. It is demonstrated that use of integer variables is completely avoided by iteratively solving a binary integer linear programming problem. This appears to be more efficient than solving the corresponding integer linear programming problem. It is demonstrated, on the basis of some cases, that this method might provide the exact solution of problems with increased complexity compared to those discussed so far in the literature. This appears particularly useful for evaluating heuristic and metaheuristic approaches.     

Novel Approaches for Analyzing Biological Networks

This paper proposes clique relaxations to identify clusters in biological networks. In particular, the maximum n-clique and maximum n-club problems on an arbitrary graph are introduced and their recognition versions are shown to be NP-complete. In addition, integer programming formulations are proposed and the results of sample numerical experiments performed on biological networks are reported.     

Tracking and outperforming large stock-market indices

Enhanced index-tracking funds aim to achieve a small target excess return over a given financial benchmark index with minimum additional risk relative to this index, i.e., a minimum tracking error. These funds are attractive to investors, especially when the index is large and thus well diversified. We consider the problem of determining a portfolio for an enhanced index-tracking fund that is benchmarked against a large stock-market index subject to real-life constraints that may be imposed by investors, stock exchanges, or investment guidelines. In the literature, various solution approaches have been proposed to enhanced index tracking that are based on different linear and quadratic tracking-error functions. However, it remains an open question which tracking-error function should be minimized to determine good enhanced index-tracking portfolios. Moreover, the existing approaches may neglect real-life constraints such as the minimum trading values imposed by stock exchanges or may not devise good feasible portfolios within a reasonable computational time when the index is large. To overcome these shortcomings, we propose novel mixed-integer linear and quadratic programming formulations and novel matheuristics. To address the open question, we minimize different tracking-error functions by applying the proposed matheuristics and exact solution approaches based on the proposed mixed-integer programming formulations in a computational experiment using a set of problem instances based on large stock-market indices with up to more than 9,000 constituents. The results of our study suggest that minimizing the so-called tracking error variance, which is a quadratic function, is preferable to minimizing other tracking-error functions.     

Mixed integer (0–1) fractional programming for decision support in paper production industry

This paper presents an effective and efficient method for solving a special class of mixed integer fractional programming (FP) problems. We take a classical reformulation approach for continuous FP as a starting point and extend it for solving a more general class of mixed integer (0–1) fractional programming problems.To stress the practical relevance of the research we focus on a real-life application in paper production industry. The constantly advancing physical knowledge of large scale pulp and paper production did have a substantial impact on an existing DSS in which mixed integer (0–1) fractional programming is introduced. We show that the motivation to solve a real-life fractional programming problem can provide the basis for a new approach in a new context that has an added value of its own, even outside the given application area. We describe the main characteristics of the DSS, the necessity to develop a non-iterative solution procedure and demonstrate both the effectiveness and efficiency of the proposed approach from practical data sets.     

A Two-Phase Exact Algorithm for MAX-SAT and Weighted MAX-SAT Problems

We describe a two-phase algorithm for MAX-SAT and weighted MAX-SAT problems. In the first phase, we use the GSAT heuristic to find a good solution to the problem. In the second phase, we use an enumeration procedure based on the Davis-Putnam-Loveland algorithm, to find a provably optimal solution. The first heuristic stage improves the performance of the algorithm by obtaining an upper bound on the minimum number of unsatisfied clauses that can be used in pruning branches of the search tree.We compare our algorithm with an integer programming branch-and-cut algorithm. Our implementation of the two-phase algorithm is faster than the integer programming approach on many problems. However, the integer programming approach is more effective than the two-phase algorithm on some classes of problems, including MAX-2-SAT problems.     

Job scheduling with dual criteria and sequence-dependent setups: mathematical versus genetic programming

Flexibility, speed, and efficiency are major challenges for operations managers in today's knowledge-intensive organizations. Such requirements are converted into three production scheduling criteria: (a) minimize the impact of setup times in flexible production lines when moving from one product to another, (b) minimize number of tardy jobs, and (c) minimize overall production time, or makespan, for a given set of products or services. There is a wide range of solution methodologies for such NP-hard scheduling problems. While mathematical programming models provide optimal solutions, they become too complex to model for large scheduling problems. Simultaneously, heuristic approaches are simpler and very often independent of the problem size, but provide “good” rather than optimal solutions. This paper proposes and compares two alternative solutions: 0-1 mixed integer linear programming and genetic programming. It also provides guidelines that can be used by practitioners in the process of selecting the appropriate scheduling methodology.     

Restarting after Branching in the SDP Approach to MAX-CUT and Similar Combinatorial Optimization Problems

Many combinatorial optimization problems have relaxations that are semidefinite programming problems. In principle, the combinatorial optimization problem can then be solved by using a branch-and-cut procedure, where the problems to be solved at the nodes of the tree are semidefinite programs. It is desirable that the solution to one node of the tree should be exploited at the child node in order to speed up the solution of the child. We show how the solution to the parent relaxation can be used as a warm start to construct an appropriate initial dual solution to the child problem. This restart method for SDP branch-and-cut can be regarded as analogous to the use of the dual simplex method in the branch-and-cut method for mixed integer linear programming problems.     

PORTFOLIO MODELING IN MULTIPLE-CRITERIA SITUATIONS UNDER UNCERTAINTY

There have been many models for portfolio selection, but most do not explicitly include uncertainty and multiple objectives. This paper presents an approach that includes these aspects using a form of stochastic integer programming with recourse. The method involves the use of a time-based decision tree structure called a “project tree.” Using this basic format, an illustrative six-project example is presented and analyzed. Various forms of objectives are discussed, ranging from the maximization of expected portfolio value to the maximization of the minimum weighted portfolio deviation from two goals. In each case, formulated numerical problems are given, and the solutions derived are presented. The approach is shown to be very flexible and capable of handling a variety of situations and objectives.     

TSP in spreadsheets—A fast and flexible tool

The traveling salesman problem (TSP) is well-known and many specially developed solution procedures have been constructed to solve particular variants of it. This paper considers several different variants of TSP. However, developing tailored solution procedures for each is impractical. These problems are non-deterministic polynomial-time hard (NP hard). Solving them using standard linear programming/mixed integer programming (LP/MIP) solvers has therefore only been regarded to be feasible for very small problems. A careful consideration of the problem formulation may facilitate efficient software utilization, and for real-world problems this can have a considerable impact. Problems that were previously regarded as large and unwieldy are now easily solvable using spreadsheets, thanks to the recent advancement in general optimization software.     

AN EXPERIMENTAL COMPARISON OF STATISTICAL AND LINEAR PROGRAMMING APPROACHES TO THE DISCRIMINANT PROBLEM

This paper reports the results of an experimental comparison of three linear programming approaches and the Fisher procedure for the discriminant problem. The linear programming approaches include two formulations proposed by Freed and Glover and a newly proposed mixed-integer, linear goal programming formulation. Ten test problems were generated for each of the 36 cells in the three-factor, full-factorial experimental design. Each test problem consisted of a 30-case estimation sample and a 1,000-case holdout sample. Experimental results indicate that each of the four approaches was statistically preferable in certain cells of the experimental design. Sample-based rules are suggested for selecting an approach based on Hotelling's T² and Box's M statistics. Subject Areas: Statistical Techniques, Linear Statistical Models, and Linear Programming.     

SCHEDULING PARALLEL PROCESSORS WITH SETUP COSTS AND RESOURCE LIMITATIONS

An integer linear programming model is presented for the scheduling of n products on m identical machines. The particular problem studied is one that occurs frequently in the fiberglass and textile industries. The model incorporates setup costs, lost production costs, and overtime costs. Due to the structure of the model, integer solutions can be obtained by explicitly restricting only a small number of the integer variables. This allows those responsible for scheduling to solve realistically sized problems in an efficient manner. Computational results are provided for a set of generated test problems.     

The triangle <Emphasis Type="Italic">k</Emphasis>-club problem

Graph models have long been used in social network analysis and other social and natural sciences to render the analysis of complex systems easier. In applied studies, to understand the behaviour of social networks and the interactions that command that behaviour, it is often necessary to identify sets of elements which form cohesive groups, i.e., groups of actors that are strongly interrelated. The clique concept is a suitable representation for groups of actors that are all directly related pair-wise. However, many social relationships are established not only face-to-face but also through intermediaries, and the clique concept misses all the latter. To deal with these cases, it is necessary to adopt approaches that relax the clique concept. In this paper we introduce a new clique relaxation—the triangle k-club—and its associated maximization problem—the maximum triangle k-club problem. We propose integer programming formulations for the problem, stated in different variable spaces, and derive valid inequalities to strengthen their linear programming relaxations. Computational results on randomly generated and real-world graphs, with \(k=2\) and \(k=3\), are reported.     

An Optimal Model for Cell Formation Decisions

This study develops an approach to cell formation decisions for cellular manufacturing layouts in group technology settings. An optimal 0–1 integer programming model is used to provide an analysis for determining which machines and parts should be assigned to cells in cellular manufacturing layouts. This approach minimizes the cost of manufacturing exceptional parts outside the cellular system, subject to machine capacity constraints. Part–machine matrices are partitioned into disconnected cells and use far fewer 0–1 variables than earlier approaches. Formulation of the model is described with a numerical example and computer solutions to realistic problems are obtained. The characteristics of computer run times, model performance, and applications of the model are discussed.     

An Arborescent Network Formulation and Dual Ascent Based Procedure for the Two-Stage Multi-Item Dynamic Demand Lotsize Problem

Traditional approaches for modeling and solving dynamic demand lotsize problems are based on Zangwill's single-source network and dynamic programming algorithms. In this paper, we propose an arborescent fixed-charge network (ARBNET) programming model and dual ascent based branch-and-bound procedure for the two-stage multi-item dynamic demand lotsize problem. Computational results show that the new approach is significantly more efficient than earlier solution strategies. The largest set of problems that could be solved using dynamic programming contained 4 end items and 12 time periods, and required 475.38 CPU seconds per problem. The dual ascent algorithms averaged .06 CPU seconds for this problem set, and problems with 30 end items and 24 time periods were solved in 85.65 CPU seconds. Similar results verify the superiority of the new approach for handling backlogged demand. An additional advantage of the algorithm is the availability of a feasible solution, with a known worst-case optimality gap, throughout the problem-solving process.