dynamic programming

Let $G=(V,E)$ be a graph.  A double Roman dominating function  (DRDF) of   $G $   is a function   $f:Vto {0,1,2,3}$  such that, for each $vin V$ with $f(v)=0$,  there is a vertex $u $  adjacent to $v$  with $f(u)=3$ or there are vertices $x$ and $y $  adjacent to $v$  such that  $f(x)=f(y)=2$ and for each $vin V$ with $f(v)=1$,  there is a vertex $u $    adjacent to $v$    with  $f(u)>1$.  The weight of a DRDF $f$ is   $ f (V) =sum_{ vin V} f (v)$.   Let $n$ and  $k$ be integers such that  $3leq 2k+ 1  leq n$.  The   generalized Petersen graph $GP (n, k)=(V,E) $  is the  graph  with  $V={u_1, u_2,ldots,  u_n}cup{v_1, v_2,ldots, v_n}$ and $E={u_iu_{i+1}, u_iv_i, v_iv_{i+k}:  1 leq i leq n}$, where  addition is taken  modulo $n$. In this paper,  we firstly   prove that the  decision     problem  associated with   double Roman domination is NP-omplete even restricted to planar bipartite graphs with maximum degree at most 4.  Next, we   give  a dynamic programming algorithm for  computing a minimum DRDF (i.e., a  DRDF   with minimum weight  along  all   DRDFs)  of $GP(n,k )$  in $O(n81^k)$ time and space  and so a  minimum DRDF  of $GP(n,O(1))$  can be computed in $O( n)$ time and space.

In this paper, we aim to propose a new hybrid version of the Longstaff and Schwartz algorithm under the exponential Levy Jump-diffusion model using Random Forest regression. For this purpose, we will build the evolution of the option price according to the number of paths. Further, we will show how this approach numerically depicts the convergence of the option price towards an equilibrium price when the number of simulated trajectories tends to a large number. In the second stage, we will compare this hybrid model with the classical model of the Longstaff and Schwartz algorithm (as a benchmark widely used by practitioners in pricing American options) in terms of computation time, numerical stability and accuracy. At the end of this paper, we will test both approaches on the Microsoft share “MSFT” as an example of a real market.

We use some recent developments in Dynamics Programming Method to obtain a rigorous solution of the epidemic model formulated in E. Trélat [Controle optimal: théorie et applications, (online version 2020)] as an unsolved problem. In fact, this problem is proposed in the context of using Pontryagin’s Maximum Principle. We use a certain refinement of Cauchy’s Method of characteristics for stratified Hamilton-Jacobi equations to describe a large set of admissible trajectories and identify a domain on which the value function exists and is generated by a certain admissible control. The optimality is justified by using of one of the well-known verification theorems as an argument for sufficient optimality conditions.

The order picking problem is one of the key elements in warehouse management.  The challenge increases during the new norm when orders can be made by going to the shop and also via online that results in high uncertainty in order volume. Despite that, customer expectation remains on fast delivery which requires the selling organizations to be able to provide fast and efficient service to meet the demand from customers.  In achieving this, among the contributing factors is efficient warehouse management especially in order picking, storage assignment, sufficient resource allocation, adequate manpower handling and proper tasking allocation. Thus, in this paper, a model for order picking is modified by considering the limited picking capacity of the Order Pickers (OP), the S-shaped route in the warehouse plan and the need for complete order (all demanded items are picked). The modified model is adapted as a Dynamic Programming problem with the objective of minimizing the time taken (through minimizing distance travelled) in picking each order. The results show that testing with a set of secondary data, the modified model shows a reduction of 24.19\% in travel distance compared to using Shortest Path Problem (SPP) and Traveling Salesman Problem (TSP). At the same time, the application of the modified model using the real data shows a reduction of 11.6\% in the travelled distance as well as more quality task allocation among the OPs.

In this paper, we develop the notion of $(psi, F)$-contraction mappings introduced in cite{wad1} in $b$-metric spaces. To achieve this, we introduce the notion of generalized multi-valued $(psi, S, F)$-contraction type I mapping with respect to generalized dynamic process $D(S, T, x_0),$ generalized multi-valued $(psi, S, F)$-contraction type II mapping with respect to generalized dynamic process $D(S, T, x_0),$ and establish common fixed point results for these classes of mappings in complete $b$-metric spaces. As an application, we obtain the existence of solutions of dynamic programming and integral equations. The results presented in this paper extends and complements some related results in the literature.

The aim of this paper is to prove a nonunique common fixed point theorem of Rhoades type for two self-mappings in complete b-metric spaces. This theorem extends the results of [17] and [49]. Examples are furnished to illustrate the validity of our results. We apply our theorem to establish the existence of common solutions of a system of two nonlinear integral equations and a system of two functional equations arising in dynamic programming.

The Iraqi Ministry of Defense decided that there would be no escape for the terrorist gang members from the death which it proceeded to it with a false doctrine until it is eradicated from Iraq which can never be an incubator of terrorism, accordingly and with the capabilities available to the ministry, the ministry has developed two strategies: The first is allocating ten regiments from the Iraqi Special Operations Forces (ISOF) to maximize the performance of the Iraqi armed forces (IAF) in defending the homeland and maintaining its security and stability from terrorist gangs in four border regions in Iraq. The second is to reduce the arrival time of the ISOF to the battlefield by determining the optimal possible paths using an efficient scientific approach which is the dynamic programming (DP). The results of this study after solving a real-life problem proved that the proposed approach is an effective mathematical approach for taking a series of related decisions by finding maximization level of performance of ISOF and the shortest time for them to reach the battlefield.