The second part of the paper presents finite-dimensional models of linear elastic, elastic-strain hardening, elastic-perfectly plastic

and rigid-perfectly plastic structures. These models can be seen as a result of discretisation procedure applied to the models of solids derived in the Part I. The implications of sub-dividing degrees of freedom into those with prescribed external forces and those with given displacements are discussed. It is pointed out that the dual energy principles given in this part of the paper can serve as a direct basis for numerical computations.

This paper presents revised and extended version of theory proposed in the late 1970-ties by A. ˇCyras and his co-workers. This theory, based upon the notion of duality in mathematical programming, allows us to generate variational principles and to investigate existence and uniqueness of solutions for the broad class of problems of elasticity and plasticity. The paper covers analysis of solids made of linear elastic, elastic-strain hardening, elastic-perfectly plastic and rigid-perfectly plastic material. The novelty with respect to ˇCyras’s theory lies in taking into account loads dispersed over the volume and displacements enforced on the part of surface. A new interpretation of optimum load for a rigid-perfectly plastic body is also given.

The issue of transportation is a particular type of mathematical programming that facilitates searching for and determining an optimal distribution network, considering the set of suppliers and recipients. This paper uses a numerical example to present a solution to a transport problem utilizing classical computation methods, i.e., the northwest corner, the least cost in a matrix, and the VAM approximation method. The objective of the paper was to develop tools in the form of algorithms that would then be implemented in three various computing environments (R, GNU Octave, and Matlab) that allow us to optimize transport costs within an assumed supply network. The model involved determining decision variables and indicating limiting conditions. Furthermore, the authors interpreted and visualized the obtained results. The implementation of the proposed solution enables users to determine an optimal transport plan for individually defined criteria.

Major manufactures are moving towards a sustainability goal. This paper introduces the results of collaboration with the leading company in the packaging and advertising industry in Germany and Poland. The problem addresses the manufacturing planning problem in terms of minimizing the total cost of production. The challenge was to bring a new production planning method into cardboard manufacturing and paper processing which minimizes waste, improves the return of expenses, and automates daily processes heavily dependent on the production planners’ experience. The authors developed a module that minimizes the total cost, which reduces the overproduction and is used by the company’s manufacturing planning team. The proposed approach incorporates planning allowances rules to compromise the manufacturing requirements and production cost minimization.

In the logistics center (warehouse or distribution center), customer orders need to be picked up by the pickers. In this research, we examine the order-picking problem with sequencedependent constraints with two decision variables (container start time and product quantity) in a distribution center with a one-directional conveyor. The decision-making is based on the developed two variations of two-step matheuristics. At first, the main order-picking problem is divided into two subproblems. Next, each step of each variant of the subproblem is solved using a mathematical programming-based technique. Both matheuristics were better in 85 of 120 test instances compared to the initial model solved by mathematical programming. Pickers matheuristics were better on average at 46.56%, while Buffers matheuristics were better on average at 46.87%. The proposed matheuristics approach allows distributors to schedule orders in the logistics center fast enough and with fewer resources.