The travelling salesman problem (TSP) is one of combinatorial optimization problems of huge importance to practical applications. However, the TSP in its “pure” form may lack some essential issues for a decision maker—e.g., time-dependent travelling conditions. Among those shortcomings, there is also a lack of possibility of not visiting some nodes in the network—e.g., thanks to the existence of some more cost-efficient means of transportation. In this article, an extension of the TSP in which some nodes can be skipped at the cost of penalties for skipping those nodes is presented under a new name and in a new mathematical formulation. Such an extension can be applied as a model for transportation cost reduction due to the possibility of outsourcing deliveries to some nodes in a TSP route. An integer linear programming formulation of such a problem based on the Gavish–Graves-flow-based TSP formulation is introduced. This formulation makes it possible to solve the considered problem by using any integer linear programming optimization software. Numerical examples and opportunities for further research are presented.

The problem of scheduling pumps is widely discussed in the literature in the context of improving energy efficiency, production costs, emissions, and reliability. In some studies, the authors analyze the available case studies and compare the results; others present their own computational methods. In the paper, a problem of pump scheduling in regular everyday operations of a water supply operator is considered. The issues of water production optimization and energy savings are part of the topic of sustainable development. The objective of the article is the minimization of the cost of electric power used by the pumps supplying water. It is achieved thanks to the variability of both the demand for water and the price of electric power during the day combined with the possibility of storing water. The formulation of an existing electric power cost optimization problem as a binary linear programming problem was improved. An essential extension of the above mathematical model, which enables more flexible management of the pump system, was also proposed. An example containing real-world input data was successfully solved using Microsoft Excel with a free OpenSolver add-in.

Purpose: This paper aims to introduce a model of fertilizer purchase optimization -an improvement of the one originally developed for supporting African farmers. The improvement takes into account necessity of purchasing fertilizers in bags of fixed weight instead of arbitrary amounts. Design/Methodology/Approach: A fertilizer purchase optimization model expressed as a nonlinear programming problem and its implementation in Microsoft Excel, once developed for a project named Optimized Fertilizer Recommendations in Africa (OFRA), were analysed. An extension of the above model in the form a mixed integer nonlinear programming problem and its implementation in Microsoft Excel were developed. Findings: The model of fertilizer purchase optimization developed for OFRA omits an important issueavailability of fertilizers in fixed-sized "portions" only (50 kg bags). An improved model which includes the inevitable purchases of fixed-sized "portions" of fertilizers into the optimality criterion is introduced. Practical Implications: The improved model is much more compliant with the conditions of the fertilizer market than the original one whereas performing the optimization remains unchanged from the point of view of the user. Originality/Value: Creating a fertilizer purchase optimization model taking into account real market conditions (sale of fertilizers in fixed-sized "portions) handles an issue which is disregarded in many existing models despite its influence on the final financial output.