Mathematical Multi-Objective Optimization of the Tactical Allocation of Machining Resources in Functional Workshops

In the aerospace industry, efficient management of machining capacity is crucial to meet the required service levels to customers and to maintain control of the tied-up working capital. We introduce new multi-item, multi-level capacitated resource allocation models with a medium--to--long--term planning horizon. The model refers to functional workshops where costly and/or time- and resource-demanding preparations (or qualifications) are required each time a product needs to be (re)allocated to a machining resource. Our goal is to identify possible product routings through the factory which minimize the maximum excess resource loading above a given loading threshold while incurring as low qualification costs as possible and minimizing the inventory.In Paper I, we propose a new bi-objective mixed-integer (linear) optimization model for the Tactical Resource Allocation Problem (TRAP). We highlight some of the mathematical properties of the TRAP which are utilized to enhance the solution process. In Paper II, we address the uncertainty in the coefficients of one of the objective functions considered in the bi-objective TRAP. We propose a new bi-objective robust efficiency concept and highlight its benefits over existing robust efficiency concepts. In Paper III, we extend the TRAP with an inventory of semi-finished as well as finished parts, resulting in a tri-objective mixed-integer (linear) programming model. We create a criterion space partitioning approach that enables solving sub-problems simultaneously. In Paper IV, using our knowledge from our previous work we embarked upon a task to generalize our findings to develop an approach for any discrete tri-objective optimization problem. The focus is on identifying a representative set of non-dominated points with a pre-defined desired coverage gap

Mathematical Optimization of the Tactical Allocation of Machining Resources in Aerospace Industry

In the aerospace industry, efficient management of machining capacity is crucial to meet the required service levels to customers (which includes, measures of quality and production lead-times) and to maintain control of the tied-up working capital. We introduce a new multi-item, multi-level capacitated planning model with a medium-to-long term planning horizon. The model can be used by most companies having functional workshops where costly and/or time- and resource demanding preparations (or qualifications) are required each time a product needs to be (re)allocated to a machining resource. Our goal is to identify possible product routings through the factory which minimizes the maximum excess resource loading above a given loading threshold, while incurring as low qualification costs as possible. In Paper I (Bi-objective optimization of the tactical allocation of jobtypes to machines), we propose a new bi-objective mathematical optimization model for the Tactical Resource Allocation Problem (TRAP). We highlight some of the mathematical properties of the TRAP which are utilized to enhance the solution process. Another contribution is a modified version of the bi-directional $\epsilon$ -constraint method especially tailored for our problem. We perform numerical tests on industrial test cases generated for our class of problem which indicates computational superiority of our method over conventional solution approaches. In Paper II (Robust optimization of a bi-objective tactical resource allocation problem with uncertain qualification costs), we address the uncertainty in the coefficients of one of the objective functions considered in the bi-objective TRAP. We propose a new bi-objective robust efficiency concept and highlight its benefits over existing robust efficiency concepts. We also suggest a solution approach for identifying all the relevant robust efficient (RE) solutions. Our proposed approach is significantly faster than an existing approach for robust bi-objective optimization problems

Bi-objective optimization of the tactical allocation of job types to machines: mathematical modeling, theoretical analysis, and numerical tests

We introduce a tactical resource allocation model for a large aerospace engine system manufacturer aimed at long-term production planning. Our model identifies the routings a product takes through the factory, and which machines should be qualified for a balanced resource loading, to reduce product lead times. We prove some important mathematical properties of the model that are used to develop a heuristic providing a good initial feasible solution. We propose a tailored approach for our class of problems combining two well-known criterion space search algorithms, the bi-directional ε-constraint method and the augmented weighted Tchebycheff method. A computational investigation comparing solution times for several solution methods is presented for 60 numerical\ua0instances

Robust optimization of a bi‑objective tactical resource allocation problem with uncertain qualification costs

In the presence of uncertainties in the parameters of a mathematical model, optimal solutions using nominal or expected parameter values can be misleading. In practice, robust solutions to an optimization problem are desired. Although robustness is a key research topic within single-objective optimization, little attention is received within multi-objective optimization, i.e. robust multi-objective optimization. This work builds on recent work within robust multi-objective optimization and presents a new robust efficiency concept for bi-objective optimization problems with one uncertain objective. Our proposed concept and algorithmic contribution are tested on a real-world\ua0multi-item capacitated resource planning\ua0problem, appearing at a large aerospace company manufacturing high precision engine parts. Our algorithm finds all the robust efficient solutions required by the decision-makers in significantly less time than the approach of Kuhn et al. (Eur J Oper Res 252(2):418–431, 2016) on 28 of the 30 industrial instances

A criterion space decomposition approach to generalized tri-objective tactical resource allocation

We present a tri-objective mixed-integer linear programming model of the tactical resource allocation problem with inventories, called the\ua0generalized tactical resource allocation problem\ua0(GTRAP). We propose a specialized criterion space decomposition strategy, in which the projected two-dimensional criterion space is partitioned and the corresponding sub-problems are solved in parallel by application of the\ua0quadrant shrinking method\ua0(QSM) (Boland in Eur J Oper Res 260(3):873–885, 2017) for identifying non-dominated points. To obtain an efficient implementation of the parallel variant of the QSM we suggest some modifications to reduce redundancies. Our approach is tailored for the GTRAP and is shown to have superior computational performance as compared to using the QSM without parallelization when applied to industrial instances

Optimization of the Offshore Wind Inter-Array Cable Layout problem using heuristic based algorithms

The current work presents an in-exact solution method used to identify feasible, and less costly inter-array cable layout for offshore wind farms. The solution method developed has been built considering the interests of wind farm developers in mind, and to support them in the planning of large offshore wind projects. The objective of the current study is to develop a fast heuristic based algorithm able to find good (less costly), feasible solution, with a small optimality gap. We are given the positions of the turbines, obstacles, and substations. The optimization problem is to find a cable layout such that there is a unique path from each turbine to one of the substations. All the turbines are connected in a series connection on a cable having a pre-defined capacity limitation. There are few additional constraints such as to prevent two or more cables from crossing each other, and cables from entering any restricted areas in the sea bed. This problem is quite similar to the wellknown Capacitated Minimum Spanning Tree (CMST) problem. The cable layout problem has been proved to be NP hard, thus, an exact algorithm is likely to have a running time that is an exponential function of the size of the input. Most of the available exact models require fast computers, and hours of computation time to find an optimal solution and still, in large instances of the problem, an optimal solution is not achieved. Although our heuristic does not guarantee an optimal solution, it has the ability to reveal good, feasible solutions in short time frame for large as well as small instances. We have implemented the heuristic in Java, and used in-built as well as customized data structures for improving the running time of the algorithm. We have tested our solution method on 8 real wind farm instances with total number of turbines ranging from 30 to 160. We have compared the results of our heuristic with the optimal solutions available for 4 wind farm instances. We achieved near optimal solutions (<1%) in most of the instances. The solution method includes a construction heuristic, which is a modified version of a well established greedy heuristic for CMST problem called Esau Williams’ heuristic. We have adapted this heuristic by introducing a procedure to find a crossing free layout, and also used a shape factor in the heuristic function to improve the solution quality. We have utilized a multiple node exchange neighborhood structure, and used it in a local search framework. The local search method improves the solution quality, and finds locally optimal solutions. In the end , we have used the algorithm on some of the larger instances with more than 100 turbines. These instances are practically impossible to solve using exact methods. We have compared our heuristic results in these instances with the actual installed layout

Mathematical optimization of the tactical allocation of machining resources for an efficient capacity utilization in aerospace component manufacturing

In the aerospace industry, with low volumes and many products, there is a critical need to efficiently use available manufacturing resources. Currently, at GKN Aerospace, resource allocation decisions that in many cases will last for several years are to some extent made with a short-term focus so as to minimize machining time, which results in a too high load on the most capable machines, and too low load on the less capable ones. This creates an imbalance in capacity utilization that leads to unnecessary queuing at some machines, resulting in long lead times and in an increase in tied-up capital. Tactical resource allocation on the medium to long-range planning horizon (six months to several years) aims to address this issue by allocating resources to meet the predicted future demand as effectively as possible, in order to ensure long range profitability. Our intent is to use mathematical optimization to find the best possible allocations

A decision-making tool to identify routings for an efficient utilization of machining resources: the decision makers’ perspective

In the aerospace industry, efficient management of machining capacity is crucial to meet the required service levels to customers (which includes, measures of quality and lead-times ) and keeps the tied-up working capital in check. The proposed decision-making tool, described in this paper, aims to combine information and knowledge of manufacturing and logistics experts in a company to improve flow of materials through the factory. The material flow situation is different for a large aerospace tier-1 supplier as opposed to flow-based manufacturing company; when there is no pandemic or natural calamity, having relatively stable demand due to long-term contract is common, but there exists short-term demand variability. There is a complex flow of products at GKN Aerospace, as the products share machining resources, thus, resulting in uneven loads at machines and sometimes excess loading at certain machines. This along with short-term demand variability results in long queues in-front of machines which contributes with the biggest share of the total lead-time. Thus, long waiting times at one/many machine/s is common and may lead to bottlenecks in many places in the production pipeline. So, there is potential benefit in having rerouting-flexibility for products which can help in reducing queuing. However, qualifying a product for a new machine is time-consuming activity, and thus, should be done few years in advance.We propose a mathematical model aimed at improving some of these deficiencies of commonly used methods by facilitating balanced resource loading levels, i.e. to provide more degrees of freedom to the planner to absorb demand variations. The output provided by the model includes production routings in each time period (quarter) for the next 4–5 years; new qualifications to be done by technical staff for allocation of part types/products to machines which are not yet qualified/used for a given product. We keep the resource loading levels that are above a given threshold as low as possible and reduce the time/money spent for qualifying new allocations