Scalarization and convergence in unified set optimization

This paper deals with scalarization and stability aspects for a unified set optimization problem. We provide characterizations for a unified preference relation and the corresponding unified minimal solution in terms of a generalized oriented distance function of the sup-inf type. We establish continuity of a function associated with the generalized oriented distance function and provide an existence result for the unified minimal solution. We establish Painlevé-Kuratowski convergence of minimal solutions of a family of scalar problems to the minimal solutions of the unified set optimization problem.

Randomized Problem-Relaxation Solving for Over-Constrained Schedules

Optimal production planning in the form of job shop scheduling problems (JSSP) is a vital problem in many industries. In practice, however, it can happen that the volume of jobs (orders) exceeds the production capacity for a given planning horizon. A reasonable aim in such situations is the completion of as many jobs as possible in time (while postponing the rest). We call this the Job Set Optimization Problem (JOP). Technically, when constraint programming is used for solving JSSPs, the formulated objective in the constraint model can be adapted so that the constraint solver addresses JOP, i.e., searches for schedules that maximize the number of timely finished jobs. However, also highly specialized solvers which proved very powerful for JSSPs may struggle with the increased complexity of the reformulated problem and may fail to generate a JOP solution given practical computation timeouts. As a remedy, we suggest a framework for solving multiple randomly modified instances of a relaxation of the JOP, which allows to gradually approach a JOP solution. The main idea is to have one module compute subset-minimal job sets to be postponed, and another one effectuating that random job sets are found. Different algorithms from literature can be used to realize these modules. Using IBM’s cutting-edge CP Optimizer suite, experiments on well-known JSSP benchmark problems show that using the proposed framework consistently leads to more scheduled jobs for various computation timeouts than a standalone constraint solver approach.

OPTIMIZATION OF PARAMETERS OF GENERALIZED DISPERSION ALGORITHM AT STATISTICAL PROCESS CONTROL

When monitoring a real production process using statistical methods, the question of early detection of violations arises. In most cases, several indicators are monitored simultaneously in the production process, and a change in the values of some indicators leads to a change in others. If there is a dependence of indicators for their monitoring, multivariate statistical control tools are used, in particular generalized variance chart. By varying the parameters of the chart, its efficiency can be significantly increased, this allows minimizing the time the process is in an unstable state.Applying the approach of A. Duncan, which he developed for Shewhart charts, a formula for the expectation of the duration of an unstable state of a process was obtained and a Python program was developed to minimize it. To test the set optimization problem, the calculation of the data of two process indicators is given and the optimal parameters of the generalized variance chart are obtained, at which the duration of the process in an unstable state is minimal.

DETERMINATION OF ACCEPTABLE RISK PARAMETERS FOR MATERIAL COSTS TO ENSURE THE QUALITY OF ENGINEERING PRODUCTS

The article presents the results of research on analog methods of risk assessment. One of the variants of these methods has been developed as a tool for practical forecasting of the optimal level of material costs to ensure a given product quality. The theoretical basis for solving this problem was the provisions of the risk management theory of technical systems. The analog method of solving the set optimization problem presented in this paper confirms the principle possibility of predicting the level of acceptable risk of material costs for organizational and technical perfection of production, taking into account consumer requirements for its quality indicators.

Scalarization and well-posedness for set optimization using coradiant sets

The aim of this paper is to study scalarization and well-posedness for a set-valued optimization problem with order relations induced by a coradiant set. We introduce the notions of the set criterion solution for this problem and obtain some characterizations for these solutions by means of nonlinear scalarization. The scalarization function is a generalization of the scalarization function introduced by Khoshkhabar-amiranloo and Khorram. Moveover, we define the pointwise notions of LP well-posedness, strong DH-well-posedness and strongly B-well-posedness for the set optimization problem and characterize these properties through some scalar optimization problem based on the generalized nonlinear scalarization function respectively.