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

Simultaneous scheduling of preventive system maintenance and of the maintenance workshop

While a system operates, its components deteriorate and in order for the system to remain operational, maintenance of its components is required. Preventive maintenance (PM) is performed so that component failure is avoided. This research aims at scheduling PM activities for a multi-component system within a finite horizon. The system to be maintained possesses positive economic dependencies, meaning that each time any component maintenance activity is performed, a common set-up cost is generated. Each component PM activity generates a cost, including replacement, service, and spare parts costs. We start from a 0-1 mixed integer linear optimization model of the PM scheduling problem with interval costs, which is to schedule PM of the components of a system over a finite and discretized time horizon, given common set-up costs and component costs, of which the latter vary with the maintenance interval. We extend the PMSPIC model to incorporate the flow of components through the maintenance/repair workshop, including stocks of spare components, both the components that require repair and the repaired ones. Our resulting model is a tight integration of the PM and the maintenance workshop scheduling. We investigate two different contract types between stakeholders, present and analyze preliminary numerical results obtained

Simultaneous scheduling of replacement and repair of common components in operating systems. A multi-objective mathematical optimization model

In order for a system to stay operational, its components need maintenance. We consider two stakeholders-a system operator and a maintenance workshop-and a contract governing their joint activities. Components in the operating systems that are to be maintained are sent to the maintenance workshop, which should perform all maintenance activities on time in order to satisfy the contract. The maintained components are then sent back to be used in the operating systems. Our modeling of this system-of-systems includes stocks of damaged and repaired components, the workshop scheduling, and the planning of preventive maintenance for the operating systems. Our modeling is based on a mixed-binary linear optimization (MBLP) model of a preventive maintenance scheduling problem with so-called interval costs over a finite and discretized time horizon. We generalize and extend this model with the flow of components through the workshop, including the stocks of spare components. The resulting scheduling model-a mixed-integer optimization (MILP) model-is then utilized to optimize the main contract in a bi-objective setting: maximizing the availability of repaired (or new) components and minimizing the costs of maintaining the operating systems over the time horizon. We analyze the main contract and briefly discuss a turn-around time contract. Our results concern the effect of our modeling on the levels of the stocks of components over time, in particular minimizing the risk for lack of spare components

Management of Wind Power Variations in Electricity System Investment Models. A Parallel Computing Strategy

Accounting for variability in generation and load and strategies to tackle variability cost-efficiently are key components of investment models for modern electricity systems. This work presents and evaluates the Hours-to-Decades (H2D) model, which builds upon a novel approach to account for strategies to manage variations in the electricity system covering several days, the variation management which is of particular relevance to wind power integration. The model discretizes the time dimension of the capacity expansion problem into 2-week segments, thereby exploiting the parallel processing capabilities of modern computers. Information between these segments is then exchanged in a consensus loop. The method is evaluated with regard to its ability to account for the impacts of strategies to manage variations in generation and load, regional resources and trade, and inter-annual linkages. Compared to a method with fully connected time, the proposed method provides solutions with an increase in total system cost of no more than 1.12%, while reducing memory requirements to 1/26’th of those of the original problem. For capacity expansion problems concerning two regions or more, it is found that the H2D model requires 1–2% of the calculation time relative to a model with fully connected time when solved on a computer with parallel processing capability

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

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

The stochastic opportunistic replacement problem, part III: improved bounding procedures

We consider the problem to find a schedule for component replacement in a multi-component system, whose components possess stochastic lives and economic dependencies, such that the expected costs for maintenance during a pre-defined time period are minimized. The problem was considered in Patriksson et al. (Ann Oper Res 224:51–75, 2015), in which a two-stage approximation of the problem was optimized through decomposition (denoted the optimization policy). The current paper improves the effectiveness of the decomposition approach by establishing a tighter bound on the value of the recourse function (i.e., the second stage in the approximation). A general lower bound on the expected maintenance cost is also established. Numerical experiments with 100 simulation scenarios for each of four test instances show that the tighter bound yields a decomposition generating fewer optimality cuts. They also illustrate the quality of the lower bound. Contrary to results presented earlier, an age-based policy performs on par with the optimization policy, although most simple policies perform worse than the optimization policy

Efficient solution of many instances of a simulation-based optimization problem utilizing a partition of the decision space

This paper concerns the solution of a class of mathematical optimization problems with simulation-based objective functions. The decision variables are partitioned into two groups, referred to as variables and parameters, respectively, such that the objective function value is influenced more by the variables than by the parameters. We aim to solve this optimization problem for a large number of parameter settings in a computationally efficient way. The algorithm developed uses surrogate models of the objective function for a selection of parameter settings, for each of which it computes an approximately optimal solution over the domain of the variables. Then, approximate optimal solutions for other parameter settings are computed through a weighting of the surrogate models without requiring additional expensive function evaluations. We have tested the algorithm\u27s performance on a set of global optimization problems differing with respect to both mathematical properties and numbers of variables and parameters. Our results show that it outperforms a\ua0standard and often applied approach based on a surrogate model of the objective function over the complete space of variables and parameters

Conditional Subgradient Methods and Ergodic Convergence in Nonsmooth Optimization

The topic of the thesis is subgradient optimization methods in convex, nonsmooth optimization. These methods are frequently used, especially in the context of Lagrangean relaxation of large scale mathematical programs where they are remarkably often able to quickly identify near-optimal Lagrangean dual solutions. We present extensions of this class of methods, insights into their theoretical properties, and numerical evaluations. The thesis consists of an introductory chapter and three research papers.In the first paper, we generalize classical subgradient optimization methods in the sense that the feasible set is taken into consideration when the step directions are determined, and establish the convergence of the resulting conditional subgradient optimization methods. A special case of these methods is obtained when the subgradient is projected onto the active constraints before the step is taken; this method is numerically evaluated in three applications, which show that its performance is significantly better than that of classical subgradient methods.In the second paper, we consider a nonsmooth, convex program solved by a conditional subgradient optimization scheme, and establish that the elements of an ergodic (averaged) sequence of subgradients in the limit fulfil the optimality conditions. This result enables the finite identification of active constraints at the solution obtained in the limit; it is also used to establish the ergodic convergence of sequences of multipliers. Further, it implies the convergence of a lower bounding procedure, thus providing a proper termination criterion for subgradient methods. Finally, we develop and establish the convergence of a simplicial decomposition scheme for nonsmooth optimization.In the third paper, we consider the application of a conditional subgradient optimization method to a Lagrangean dual formulation of a convex program. Normally, dual subgradient schemes produce neither primal feasible nor primal optimal solutions automatically. We establish that an ergodic sequence of Lagrangean subproblem solutions converges to the primal optimal set. Numerical experiments show that the primal solution thus generated are of considerably higher quality than the Lagrangean subproblem solutions produced by the subgradient scheme