Methods for many-objective optimization: an analysis

Decomposition-based methods are often cited as the solution to problems related with many-objective optimization. Decomposition-based methods employ a scalarizing function to reduce a many-objective problem into a set of single objective problems, which upon solution yields a good approximation of the set of optimal solutions. This set is commonly referred to as Pareto front. In this work we explore the implications of using decomposition-based methods over Pareto-based methods from a probabilistic point of view. Namely, we investigate whether there is an advantage of using a decomposition-based method, for example using the Chebyshev scalarizing function, over Paretobased methods

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

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

AUTOMATING DATA-LAYOUT DECISIONS IN DOMAIN-SPECIFIC LANGUAGES

A long-standing challenge in High-Performance Computing (HPC) is the simultaneous achievement of programmer productivity and hardware computational efficiency. The challenge has been exacerbated by the onset of multi- and many-core CPUs and accelerators. Only a few expert programmers have been able to hand-code domain-specific data transformations and vectorization schemes needed to extract the best possible performance on such architectures. In this research, we examined the possibility of automating these methods by developing a Domain-Specific Language (DSL) framework. Our DSL approach extends C++14 by embedding into it a high-level data-parallel array language, and by using a domain-specific compiler to compile to hybrid-parallel code. We also implemented an array index-space transformation algebra within this high-level array language to manipulate array data-layouts and data-distributions. The compiler introduces a novel method for SIMD auto-vectorization based on array data-layouts. Our new auto-vectorization technique is shown to outperform the default auto-vectorization strategy by up to 40% for stencil computations. The compiler also automates distributed data movement with overlapping of local compute with remote data movement using polyhedral integer set analysis. Along with these main innovations, we developed a new technique using C++ template metaprogramming for developing embedded DSLs using C++. We also proposed a domain-specific compiler intermediate representation that simplifies data flow analysis of abstract DSL constructs. We evaluated our framework by constructing a DSL for the HPC grand-challenge domain of lattice quantum chromodynamics. Our DSL yielded performance gains of up to twice the flop rate over existing production C code for selected kernels. This gain in performance was obtained while using less than one-tenth the lines of code. The performance of this DSL was also competitive with the best hand-optimized and hand-vectorized code, and is an order of magnitude better than existing production DSLs.Doctor of Philosoph

Deep Learning the Efficient Frontier of Convex Vector Optimization Problems

In this paper, we design a neural network architecture to approximate the weakly efficient frontier of convex vector optimization problems (CVOP) satisfying Slater's condition. The proposed machine learning methodology provides both an inner and outer approximation of the weakly efficient frontier, as well as an upper bound to the error at each approximated efficient point. In numerical case studies we demonstrate that the proposed algorithm is effectively able to approximate the true weakly efficient frontier of CVOPs. This remains true even for large problems (i.e., many objectives, variables, and constraints) and thus overcoming the curse of dimensionality