The robust bilevel continuous knapsack problem with uncertain coefficients in the follower’s objective

We consider a bilevel continuous knapsack problem where the leader controls the capacity of the knapsack and the follower chooses an optimal packing according to his own profits, which may differ from those of the leader. To this bilevel problem, we add uncertainty in a natural way, assuming that the leader does not have full knowledge about the follower’s problem. More precisely, adopting the robust optimization approach and assuming that the follower’s profits belong to a given uncertainty set, our aim is to compute a solution that optimizes the worst-case follower’s reaction from the leader’s perspective. By investigating the complexity of this problem with respect to different types of uncertainty sets, we make first steps towards better understanding the combination of bilevel optimization and robust combinatorial optimization. We show that the problem can be solved in polynomial time for both discrete and interval uncertainty, but that the same problem becomes NP-hard when each coefficient can independently assume only a finite number of values. In particular, this demonstrates that replacing uncertainty sets by their convex hulls may change the problem significantly, in contrast to the situation in classical single-level robust optimization. For general polytopal uncertainty, the problem again turns out to be NP-hard, and the same is true for ellipsoidal uncertainty even in the uncorrelated case. All presented hardness results already apply to the evaluation of the leader’s objective function.

Impact of Trapezoidal Demand and Deteriorating Preventing Technology in an Inventory Model in Interval Uncertainty under Backlogging Situation

The demand for a product is one of the important components of inventory management. In most cases, it is not constant; it may vary from time to time depending upon several factors which cannot be ignored. For any seasonal product, it is observed that at the beginning of the season, demand escalates over time, then it is stable and after that, it decreases. This type of demand is known as the trapezoidal type. Also, due to the uncertainty of customers’ behavior, inventory parameters are not always fixed. Combining these two concepts together, an inventory model is formulated for decaying items in an interval environment. Preservative technology is incorporated to preserve the product from deterioration. The corresponding mathematical formulation is derived in such a way that the profit of the inventory system is maximized. Consequently, the corresponding optimization problem is converted into an interval optimization problem. To solve the same, different variants of quantum-behaved particle swarm optimization (QPSO) techniques are employed to determine the duration of stock-in time and preservation technology cost. To illustrate and also to validate the model, three numerical examples are considered and solved. Then the computational results are compared. Thereafter, to study the impact of different parameters of the proposed model on the best found (optimal or very close to optimal) solution, sensitivity analysis are performed graphically.

Development of a model of selecting cloud software for a road construction organization under interval information

The object of research is the management processes of a road construction organization. The research is based on the principles of systems analysis for structuring design processes; methods of mathematical modeling, fuzzy mathematics, discrete programming, multicriteria assessment and optimization for the selection of cloud software for road construction organizations in terms of interval information. The information system of a road construction organization includes planning, reporting, regulatory and technical documentation that characterizes the state and movement of information in the enterprise. It is important to use systems that speed up the generation, processing and preparation of documents, as well as improve the storage and retrieval of information. The introduction of cloud technologies has become a necessary condition for increasing the mobility, flexibility and efficiency of the management system of a road construction organization. Formalized processes of information collection and internal distribution can better predict the dynamics of market trends and act more quickly, make decisions confidently and reasonably. In the final stages of selection for assessment, it is convenient to apply the criteria in conditions of interval uncertainty. The study was aimed at improving the efficiency of transport management by developing a model for choosing the cloud software of a road construction organization in terms of interval information. The following criteria of partial optimization were used in the developed model: maximum speed of execution of functions by cloud software; minimum cloud software requirements for internet connection speed; minimum cost of cloud software. The scope of permissible solutions is determined by restrictions: – execution of all functions must be provided by cloud software; – the minimum speed of execution of functions by the cloud software should be not lower than set; – cloud software requirements for Internet connection speed should not exceed the specified; – the cost of cloud software should be no more than specified. The developed model will reduce the cost of purchasing cloud software and increase the efficiency of transport management of a road construction organization.

Combined method for correction interval systems of linear algebraic equations

The possibility of application of the interval analysis for data processing in the field of spectral analysis is considered. It is assumed that the data have interval uncertainty; therefore the problem of finding unknown concentrations is posed as a linear interval tolerance problem. The incompatibility of the interval system of linear algebraic equations is shown for the initial data using the apparatus of the recognizing functional. The relevance of the topic is due to the need for regularization of inconsistent interval systems of linear equations. The idea of S. P. Shary of a combined method for correcting a linear tolerance problem has been implemented. A new method for managing the solution by changing the linear algebraic equations interval system matrix elements radii has been developed. The research results can be used for example, to calculate the substance’s concentrations by measurement of the characteristic X-ray radiation.