In the paper we present a new approach to solving \emph{NP}-hard problems of discrete optimization adapted to the architecture of quantum processors { (QPU, Quantum Processor Unit)} implementing hardware quantum annealing. This approach is based on the use of the quantum annealing metaheuristic in the exact branch and bound algorithm to compute the lower and upper bounds of the objective function. To determine the lower bound, a new method of defining the Lagrange function for the dual problem ({the} generalized discrete knapsack problem) was used, the value of which is calculated on the QPU of a quantum machine. In turn, to determine the upper bound, we formulate an appropriate task in the form of binary quadratic programming with constraints. Despite the fact that the results generated by the quantum machine are probabilistic, the hybrid method of algorithm construction proposed in the paper, using alternately {a} CPU and QPU, guarantees the optimal solution. As a case study we consider the \emph{NP}-hard single machine scheduling problem with minimizing the weighted number of tardy jobs. The performed computational experiments showed that optimal solutions were {already} obtained in the root of the solution tree, and the values of the lower and upper bounds differ by only a few percent.

The chapter is devoted to scheduling of jobs performed by machines and by an operator in the automated manufacturing cell, which produces parts in large production batches. The purpose of scheduling is to determine a cyclical schedule that minimizes production cycle time. The chapter presents the original model of the problem that enables effective determination of cycle time for any sequence of operations in the cell. What is more, there was an algorithm proposed that determines the sequence and schedule of works minimizing the production cycle time.

Streszczenie: System pracy potokowej w budownictwie dotyczy realizacji kompleksu obiektów składających się z wielu jednakowych prac wykonywanych kolejno przez wyspecjalizowane brygady. Rozpatrywany jest problem harmonogramowania projektu z niepewnymi czasami wykonywania prac reprezentowanymi przez liczby rozmyte lub rozkłady zmiennych losowych. Przedstawiamy algorytmy memetyczne (hybrydowe algorytmy genetyczne) oraz eksperymenty obliczeniowe, których celem było zbadanie stabilności wyznaczanych rozwiązań. Słowa kluczowe: problem potokowy, niepewne dane, harmonogramowanie, algorytm memetyczny.