Examples of graphs that are rainbow domination regular and not vertex transitive are given. This answers two questions asked in Kuzman (2020). We also characterize all generalized Petersen graphs ...that are 3-rainbow domination regular.
Retail supply chains operate in a constantly changing environment and need to adapt to different situations in order to increase their reliability, flexibility and convenience. Holding and ...transportation costs can amount to up to 40 per cent of the product value, so that the proper coordination of interrelated activities plays an essential role when managing retail flows. In order to provide a relevant model we first focus on future demand satisfaction, whereas pricing policies, perishability factors, etc., are subjected to a complementary model for operative planning. The idea is to obtain a preferable distribution plan with minimal expected distribution costs, as well as minimal supply risks. The used methodology produces a set of solutions and quality estimates which can be used in order to find a desired distribution plan which is near-optimal. While considering stochasticity on the demand side, a multi-objective optimisation approach is introduced to cope with the minimisation of transport and warehouse costs, the minimisation of overstocking effects and the maximisation of customer's service level. The optimisation problem that arises is a computationally hard problem. A computational experiment has shown that the version of the problem where the weighted sum of costs is minimised can be handled sufficiently well by some well-known simple heuristics.
In this paper we study sufficient matrices, which play an important role in theoretical analysis of interior-point methods for linear complementarity problems. We present new characterisations of ...these matrices which imply new necessary and sufficient conditions for sufficiency. We use these results to develop an algorithm with exponential iteration complexity which in each iteration solves a simple instance of linear programming problem and is capable to reveal whether given symmetric matrix is sufficient or not. This algorithm demonstrates 100 % accuracy on all tested instances of matrices.
A double Roman dominating function on a graph G=(V,E) is a function f:V→{0,1,2,3}, satisfying the condition that every vertex u for which f(u)=1 is adjacent to at least one vertex assigned 2 or 3, ...and every vertex u with f(u)=0 is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2. The weight of f equals the sum w(f)=∑v∈Vf(v). The minimum weight of a double Roman dominating function of G is called the double Roman domination number γdR(G) of a graph G. We obtain tight bounds and in some cases closed expressions for the double Roman domination number of generalized Petersen graphs P(ck,k). In short, we prove that γdR(P(ck,k))=32ck+ε, where limc→∞,k→∞εck=0.
New results on singleton rainbow domination numbers of generalized Petersen graphs P(ck,k) are given. Exact values are established for some infinite families, and lower and upper bounds with small ...gaps are given in all other cases.
The rapid worldwide evolution of LEDs as light sources has brought new challenges, which means that new methods are needed and new algorithms have to be developed. Since the majority of LED ...luminaries are of the multi-source type, established methods for the design of light engines cannot be used in the design of LED light engines. This is because in the latter case what is involved is not just the design of a good reflector or projector lens, but the design of several lenses which have to work together in order to achieve satisfactory results. Since lenses can also be bought off the shelf from several manufacturers, it should be possible to combine together different off the shelf lenses in order to design a good light engine. However, with so many different lenses to choose from, it is almost impossible to find an optimal combination by hand, which means that some optimization algorithms need to be applied. In order for them to work properly, it is first necessary to describe the input data (i.e. spatial light distribution) in a functional form using as few as possible parameters. In this paper the focus is on the approximation of the input data, and the implementation of the well-known mathematical procedure for the separation of linear and nonlinear parameters, which can provide a substantial increase in performance.
Double Roman Domination: A Survey Rupnik Poklukar, Darja; Žerovnik, Janez
Mathematics (Basel),
01/2023, Letnik:
11, Številka:
2
Journal Article
Recenzirano
Odprti dostop
Since 2016, when the first paper of the double Roman domination appeared, the topic has received considerable attention in the literature. We survey known results on double Roman domination and some ...variations of the double Roman domination, and a list of open questions and conjectures is provided.
We obtain new results on the 2-rainbow domination number of generalized Petersen graphs P(ck,k). Exact values are established for all infinite families where the general lower bound 45ck is attained. ...In all other cases lower and upper bounds with small gaps are given.
We obtain new results on 3-rainbow domination numbers of generalized Petersen graphs P(6k,k). In some cases, for some infinite families, exact values are established; in all other cases, the lower ...and upper bounds with small gaps are given. We also define singleton rainbow domination, where the sets assigned have a cardinality of, at most, one, and provide analogous results for this special case of rainbow domination.
PDF
