Three perspectives for solving the job grouping problem

The production efficiency of printed circuit board (PCB) assembly depends strongly on the organization of the component placement jobs. This is characteristic, especially in a high-mix low-volume production environment. The present study discusses the problem of arranging the jobs of one machine into groups in such a way that the job change costs will be minimized when the costs depend on the number of the job groups. This problem is motivated by the practical case where the group utilizes a common machine set-up and the number of set-up occasions is the dominating factor in the production line optimization. The problem is well known and its large instances are hard to solve to optimality. We show how real-life problem instances can be solved by three different methods: efficient heuristics, 0/1-programming, and constraint programming. The first two of these are standard approaches in the field, whereas the application of constraint programming is new for the job grouping problem. The heuristic approach turns out to be efficient: algorithms are fast and produce optimal or nearly optimal groupings. 0/1-programming is capable of finding optimal solutions to small problem instances and it therefore serves as a benchmark to approximative methods. The constraint approach solves moderately large problem instances to optimality and it has the great advantage that changing the problem formulation is relatively easy one can add new constraints or modify the details of the existing ones flexibly.     

Branching strategies to improve regularity of crew schedules in ex-urban public transit

We discuss timetables in ex-urban bus traffic that consist of many trips serviced every day together with some exceptions that do not repeat daily. Traditional optimization methods for vehicle and crew scheduling in such cases usually produce schedules that contain irregularities which are not desirable especially from the point of view of the bus drivers. We propose a solution method which improves regularity while partially integrating the vehicle and crew scheduling problems. The approach includes two phases: first we solve the LP relaxation of a set covering formulation, using column generation together with Lagrangean relaxation techniques. In a second phase, we generate integer solutions using a new combination of local branching and various versions of follow-on branching. Numerical tests with artificial and real instances show that regularity can be improved significantly with no or just a minor increase of costs.     

An adaptive multi-population genetic algorithm for job-shop scheduling problem

Job-shop scheduling problem (JSP) is a typical NP-hard combinatorial optimization problem and has a broad background for engineering application. Nowadays, the effective approach for JSP is a hot topic in related research area of manufacturing system. However, some JSPs, even for moderate size instances, are very difficult to find an optimal solution within a reasonable time because of the process constraints and the complex large solution space. In this paper, an adaptive multi-population genetic algorithm (AMGA) has been proposed to solve this problem. Firstly, using multi-populations and adaptive crossover probability can enlarge search scope and improve search performance. Secondly, using adaptive mutation probability and elite replacing mechanism can accelerate convergence speed. The approach is tested for some classical benchmark JSPs taken from the literature and compared with some other approaches. The computational results show that the proposed AMGA can produce optimal or near-optimal values on almost all tested benchmark instances. Therefore, we can believe that AMGA can be considered as an effective method for solving JSP.     

Lightweight design of bus frames from multi-material topology optimization to cross-sectional size optimization

A comprehensive solution for bus frame design is proposed to bridge multi-material topology optimization and cross-sectional size optimization. Three types of variables (material, topology and size) and two types of constraints (static stiffness and frequencies) are considered to promote this practical design. For multi-material topology optimization, an ordered solid isotropic material with penalization interpolation is used to transform the multi-material selection problem into a pure topology optimization problem, without introducing new design variables. Then, based on the previously optimal topology result, cross-sectional sizes of the bus frame are optimized to further seek the least mass. Sequential linear programming is preferred to solve the two structural optimization problems. Finally, an engineering example verifies the effectiveness of the presented method, which bridges the gap between topology optimization and size optimization, and achieves a more lightweight bus frame than traditional single-material topology optimization.     

Tabu search-based algorithm for the TOC product mix decision

The theory of constraints (TOC) is a management philosophy that maximizes profits in a manufacturing plant with a demonstrated bottleneck. The product mix decision is one application of TOC that involves determination of the quantity and the identification of each product to produce. However, the original TOC heuristic is considered to produce unrealizable solution when a manufacturing plant has multiple resource constraints. This paper presents a tabu search-based TOC product mix heuristic to identify optimal or near optimal product mix for small problem instances under conditions where the original TOC heuristic failed. The tabu search-based TOC product mix heuristic is further used to solve large problem instances typical of practical manufacturing scenario. The experimental results for small to medium size problem show that the tabu search-based TOC heuristic compares favourably with those of optimal methods. Large size problems for which optimal methods have not been established in terms of feasibility in computation times were also solved in reasonable times with good quality solutions, thus confirming that the proposed approach is appropriate for adoption by production planners for the product mix problem in the manufacturing industry.     

A robust optimization approach for the capacitated vehicle routing problem with demand uncertainty

In this paper we introduce a robust optimization approach to solve the Vehicle Routing Problem (VRP) with demand uncertainty. This approach yields routes that minimize transportation costs while satisfying all demands in a given bounded uncertainty set. We show that for the Miller-Tucker-Zemlin formulation of the VRP and specific uncertainty sets, solving for the robust solution is no more difficult than solving a single deterministic VRP. Our computational results on benchmark instances and on families of clustered instances show that the robust solution can protect from unmet demand while incurring a small additional cost over deterministic optimal routes. This is most pronounced for clustered instances under moderate uncertainty, where remaining vehicle capacity is used to protect against variations within each cluster at a small additional cost. We compare the robust optimization model with classic stochastic VRP models for this problem to illustrate the differences and similarities between them. We also observe that the robust solution amounts to a clever management of the remaining vehicle capacity compared to uniformly and non-uniformly distributing this slack over the vehicles.     

Cyclic and non-cyclic crew rostering problems in public bus transit

The crew rostering problem arises in public transport bus companies, and addresses the task of assigning a given set of anonymous duties and some other activities, such as standbys and days off, to drivers or groups of drivers, without violating any complex labor union rules. In addition, the preferences of drivers are considered during the assignment. The plan generated for each driver/group of drivers is called a roster. Optimal rosters are characterized by maximum satisfaction of drivers and minimal operational costs. To generate a personalized roster for each driver/group of drivers, the problem is formulated as a multi-commodity network flow problem in this paper. In each network layer, a roster is generated for each driver or driver group. The network model is very flexible and can accommodate a variety of constraints. In addition, with a minor modification, the network can formulate the cyclic and non-cyclic crew rostering problems. To the best of our knowledge, this is the first publication which solves both problems with one model. The main goal of this paper is to develop a mixed-integer mathematical optimization network model for both problems with sequential and integrated approaches and to solve this model using commercial solvers. Both problems are usually solved with the sequential approach. Therefore, another contribution of this paper is comparing the sequential approach with the integrated one. Our experiments on real-world instances show that the integrated approach outperforms the sequential one in terms of solution quality.     

On solving three-dimensional open-dimension rectangular packing problems

In this article, a recently proposed three-dimensional open-dimension rectangular packing problem is considered, in which the objective is to find a minimal volume rectangular container that packs a set of rectangular boxes. The literature has tackled small-sized instances of this problem by means of optimization solvers, position-free mixed-integer programming (MIP) formulations and piecewise linearization approaches. In this study, the problem is alternatively addressed by means of grid-based position MIP formulations, whereas still considering optimization solvers and the same piecewise linearization techniques. A comparison of the computational performance of both models is then presented, when tested with benchmark problem instances and with new instances, and it is shown that the grid-based position MIP formulation can be competitive, depending on the characteristics of the instances. The grid-based position MIP formulation is also embedded with real-world practical constraints, such as cargo stability, and results are additionally presented.     

Optimal capacity expansion for multi-product, multi-machine manufacturing systems with stochastic demand

We consider a discrete-time capacity expansion problem involving multiple product families, multiple machine types, and non-stationary stochastic demand. Capacity expansion decisions are made to strike an optimal balance between investment costs and lost sales costs. Motivated by current practices in the semiconductor and other high-tech industries, we assume that only minimal amounts of finished-goods inventories are held, due to the risk of obsolescence. We assume that when capacity is in short supply, management desires to ensure that a minimal service level for all product families is obtained. Our approach uses a novel assumption that demand can be approximated by a distribution whose support is a collection of rays emanating from a point and contained in real multi-dimensional space. These assumptions allow us to solve the problem as a max-flow, min-cut problem. Computational experiments show that large problems can be solved efficiently.     

Lagrangean relaxation and hybrid simulated annealing tabu search procedure for a two-echelon capacitated facility location problem with plant size selection

In this paper, we study a two-echelon capacitated facility location problem with plant size selection (TECFLP-PSS). Given a set of potential sites for plants, each of which is associated with several possible sizes and corresponding unit production costs, a set of potential sites for capacitated depots and a set of customers with demands, the TECFLP-PSS aims to optimise the plant locations and sizes, the depot locations and the product flows from the opened plants to the opened depots and then to the end customers under single-source constraints. The objective is to satisfy all customers’ demands with a minimum total cost of facility opening, production and transportation. We develop a mixed integer programming model and propose a Lagrangean relaxation approach combined with new valid inequalities and core problem to achieve tight lower and upper bounds for this problem. We then improve the upper bound with a hybrid simulated annealing tabu search procedure. Computational experiments on benchmarks and randomly generated instances are conducted to validate the effectiveness and efficiency of the proposed method.     

Optimal batch ordering policies for assembly systems with guaranteed service

For a supply chain modelled as a multi-echelon inventory system, effective management of its inventory at each stock is critical to reduce inventory costs while assuring a given service level to customers. In our previous work, we used the guaranteed-service approach (GSA) to design optimal echelon batch ordering policies for continuous-review serial systems with Poisson customer demand and fixed order costs. The approach assumes that the final customer demand is bounded and each stock has a guaranteed service time in the sense that the demand of its downstream stock can always be satisfied in the service time. This paper extends this work by considering more general assembly systems. We first derive an analytical expression for the total cost of the system in the long run. The problem of finding optimal echelon batch ordering policies for the system can then be decomposed into two independent sub-problems: order size decision sub-problem and reorder point decision sub-problem. We develop efficient dynamic programming algorithms for the two sub-problems. Numerical experiments on randomly generated instances show the effectiveness of the proposed approach.     

Batch sizes optimisation by means of queueing network decomposition and genetic algorithm

Batch sizes have a considerable impact on the performance of a manufacturing process. Determining optimal values for batch sizes helps to reduce inventories/costs and lead times. The deterministic nature of the available batch size optimisation models reduces the practical value of the obtained solutions. Other models focus only on critical parts of the system (e.g., the bottleneck). In this paper, we present an approach that overcomes important limitations of such simplified solutions. We describe a combination of queueing network analysis and a genetic algorithm that allows us to take into account the real characteristics of the system when benefiting from an efficient optimisation mechanism. We are able to demonstrate that the application of our approach on a real-sized problem with 49 products allows us to obtain a solution (values for batch sizes) with less than 4% relative deviation of the cycle time from the exact minimal value.     

Air cargo scheduling: integrated models and solution procedures

Designing a profitable flight schedule is a highly complex planning problem. Both passenger and cargo airlines usually follow a decomposition approach and break this problem into several subproblems which are then solved consecutively and iteratively using specific but isolated models. At cargo airlines, the four major interdependent decision problems are flight selection, fleet assignment, rotation planning, and cargo routing. In our research, we have developed a planning approach which differs from other OR-based planning approaches in two aspects. The approach is based on integrated models and it is based on the pragmatic planning paradigm to optimally modify an existing schedule. For this purpose, the planner has to identify mandatory and optional flights. Then the planning goal is to identify the best combination of optional flights to be included into the schedule. Our integrated planning models comprise several additional important planning aspects for cargo airlines such as available capacities on external flights (e.g. belly capacities from passenger flights or road-feeder services), cargo handling costs and constraints, and aircraft maintenance regulations. There are two main aspects which we present in this paper. First, we describe the planning problem and the specific planning paradigm, develop a set of complex mixed-integer programs representing the different subproblems, and finally present integrated problem formulations as well as several model extensions. Thereafter, we develop a branch and price and cut approach for solving the mathematical programs and present extensive computational results obtained for a set of generated yet highly practical problem instances for different types of carriers. The results show that our approach is able to find high quality solutions to problem instances of realistic size and complexity within reasonable time.     

An optimal and a heuristic algorithm for the single-item retrieval problem in puzzle-based storage systems with multiple escorts

Puzzle-based storage systems consist of densely stored unit loads on a square grid. The problem addressed in this paper is to retrieve a stored unit load from a puzzle-based storage using the minimum number of item moves. While previous research contributed optimal algorithms for only up to two empty locations (escorts), our approach solves configurations where multiple empty locations are arbitrarily positioned in the grid. The problem is formulated as a state space problem and solved to optimality using an exact search algorithm. To reduce the search space, we derive bounds on the number of eligible empty locations and develop several search-guiding estimate functions. Furthermore, we present a heuristic variant of the search algorithm to solve larger problem instances. We evaluate both solution algorithms on a large set of problem instances. Our computational results show that the algorithms clearly outperform existing approaches where they are applicate and solve more general configurations, which could not be solved to optimality before. The heuristic variant efficiently yields high-quality solutions for significantly larger instances of practically relevant size.     

Practical solutions for multi-objective optimization: An application to system reliability design problems

For multiple-objective optimization problems, a common solution methodology is to determine a Pareto optimal set. Unfortunately, these sets are often large and can become difficult to comprehend and consider. Two methods are presented as practical approaches to reduce the size of the Pareto optimal set for multiple-objective system reliability design problems. The first method is a pseudo-ranking scheme that helps the decision maker select solutions that reflect his/her objective function priorities. In the second approach, we used data mining clustering techniques to group the data by using the k-means algorithm to find clusters of similar solutions. This provides the decision maker with just k general solutions to choose from. With this second method, from the clustered Pareto optimal set, we attempted to find solutions which are likely to be more relevant to the decision maker. These are solutions where a small improvement in one objective would lead to a large deterioration in at least one other objective. To demonstrate how these methods work, the well-known redundancy allocation problem was solved as a multiple objective problem by using the NSGA genetic algorithm to initially find the Pareto optimal solutions, and then, the two proposed methods are applied to prune the Pareto set.     

An Efficient Algorithm for Global Path Optimization in MPLS Networks

This paper proposes an algorithm to solve the optimization of label switched paths (LSPs) in multiprotocol label switching (MPLS) networks. The underlying optimization problem in this task is the well-known unsplittable multicommodity flow problem equipped with practically relevant objective functions and specialized with hard technical requirements.The proposed heuristic algorithm is based on network flow theory. It incorporates iterative shortest path search and performs adaptive edge weight adjustments in order to successfully satisfy all the required traffic demands and to maximize user-defined objectives. The robust algorithm facilitates the incorporation of several strategic and optimization objectives and the fulfillment of certain hard technical requirements of the target problem domain as well. Novel features of the approach include a new adaptive path allocation/deallocation strategy based on the identification of bottleneck links, demand ordering and preprocessing phases, and a systematic path allocation control method.The efficiency of the method is empirically shown on randomly generated networks with practical sizes and topologies, and on a real-world IP (Internet Protocol) backbone network. The algorithm is able to successfully solve difficult problem instances comprising very large instances with 1000 nodes, 3500 edges and 999000 traffic demands. The computational tests demonstrate that the proposed approach can be efficiently applied to solve problem instances that embed MPLS specific hard technical requirements. Furthermore, it is shown that our algorithm offers significantly better performance than the straightforward adaptations of existing methods that were developed for related network optimization problems. Namely, our algorithm produces acceptable results quicker, it can solve problems that were not previously solvable, and it yields better results than the alternative methods. The extensive empirical tests demonstrate the combinatorial properties of the target problem and the performance aspects of the algorithm and its components as well.     

Tramp ship routing and scheduling with voyage separation requirements

In this paper we explore tramp ship routing and scheduling. Tramp ships operate much like taxies following the available demand. Tramp operators can determine some of their demand in advance by entering into long-term contracts and then try to maximise profits from optional voyages found in the spot market. Routing and scheduling a tramp fleet to best utilise fleet capacity according to current demand is therefore an ongoing and complicated problem. Here we add further complexity to the routing and scheduling problem by incorporating voyage separation requirements that enforce a minimum time spread between some voyages. The incorporation of these separation requirements helps balance the conflicting objectives of maximising profit for the tramp operator and minimising inventory costs for the charterer, since these costs increase if similar voyages are not performed with some separation in time. We have developed a new and exact branch-and-price procedure for this problem. We use a dynamic programming algorithm to generate columns and describe a time window branching scheme used to enforce the voyage separation requirements which we relax in the master problem. Computational results show that our algorithm in general finds optimal solutions very quickly and performs much faster compared to an earlier a priori path generation method. Finally, we compare our method to an earlier adaptive large neighbourhood search heuristic and find that on similar-sized instances our approach generally uses less time to find the optimal solution than the adaptive large neighbourhood search method uses to find a heuristic solution.

     

A graph partitioning strategy for solving large-scale crew scheduling problems

Railway crew scheduling deals with generating driver duties for a given train timetable such that all work regulations are met and the resulting schedule has minimal cost. Typical problem instances in the freight railway industry require the generation of duties for thousands of drivers operating tens of thousands of trains per week. Due to short runtime requirements, common solution approaches decompose the optimization problem into smaller subproblems that are solved separately. Several studies have shown that the way of decomposing the problem significantly affects the solution quality. An overall best decomposition strategy for a freight railway crew scheduling problem, though, is not known. In this paper, we present general considerations on when to assign two scheduled train movements to separate subproblems (and when to rather assign them to the same subproblem) and deduct a graph partitioning based decomposition algorithm with several variations. Using a set of real-world problem instances from a major European railway freight carrier, we evaluate our strategy and benchmark the performance of the decomposition algorithm both against a common non-decomposition algorithm and a lower bound on the optimal solution schedule. The test runs show that our decomposition algorithm is capable of producing high-quality solution schedules while significantly cutting runtimes compared to the non-decomposition solution algorithm. We are following a ”greenfield” approach, where no information on previous schedules is needed. Hence, our approach is applicable to any railway crew scheduling setting, including network enlargement, integration of new customers, etc.     

Travel distance estimation and storage zone optimization in a 2-block class-based storage strategy warehouse

Order picking has been considered as the most critical operation in warehousing. Recent trends in logistics demand faster but more reliable order picking systems. The efficiency of an order picking process greatly depends on the storage policy used, i.e. where products are located within the warehouse. In this paper, we deal with the most popular storage policy that is class-based (or ABC) storage strategy. Particularly, we investigate the problem of determining the optimal storage boundaries (zones) of classes in each aisle for manually operated warehouses.

We first propose a probabilistic model that enables us to estimate the average travel distance of a picking tour. We found that the differences between results obtained from simulation and the model were slight. Using the average travel distance as the objective function, we present a mathematical formulation for the storage zone optimization problem. However, the exact approach can handle only small size warehouse instances. To circumvent this obstacle, we propose a heuristic for the problem. Numerical examples we conducted show that the heuristic performs very well in all the cases.     

Capacitated lot-sizing with sequence dependent setup costs

In this paper we consider a single-stage system where a number of different items have to be manufactured on one machine. Expenditures for the setups depend on the sequence in which items are scheduled on the machine. Holding costs are incurred for holding items in inventory. The demand of the items has to be satisfied without delay, i.e. shortages are not allowed. The objective is to compute a schedule such that the sum of holding and setup costs is minimized with respect to capacity constraints. For this problem which we call capacitated lot-sizing problem with sequence dependent setup costs (CLSD) we formulate a new model. The main differences between the new model and the discrete lot-sizing problem with sequence dependent setup costs (DLSDSD), introduced by Fleischmann, is that continuous lot-sizes are allowed and the setup state can be preserved over idle time. For the solution of the new model we present a heuristic which applies a priority rule. Since the priority values are affected by two significant parameters, we perform a local search in the parameter space to obtain low cost solutions. The solution quality is analyzed by a computational study. The comparison with optimal solutions of small instances shows that the solution quality of our heuristic is acceptable. The Fleischmann approach for the DLSPSD computes upper bounds for our new problem. On the basis of larger instances we show that our heuristic is more efficient to solve the CLSD.