Open Access Original Research Article

Hassan Alsuhabi, Rhonda Magel

**Aims: **Introducing and comparing 4 different tests for the unknown umbrella alternative in a mixed design.

**Study Design:** Simulation study consisting of a randomized complete block portion and a completely randomized design portion for various underlying distributions.

**Place and Duration of Study:** Simulation Study – conducted at North Dakota State University from September 2018 through December 2019.

**Methodology:** This paper proposes four non-parametric tests for testing the umbrella alternative with unknown peak when the data are mixture of a randomized complete block and a completely randomized design. The proposed tests are various combinations of a modified (unmodified) Mack-Wolfe’s test and a modified (unmodified) Kim-Kim’s test, respectively. In this paper, the proposed tests are an extension of Magel et al. (2010) and Hassan and Magel (2020) peak known tests to the unknown peak setting. The four proposed test statistics are compared to each other.

**Results:** When there were 3 populations, the unmodified versions of the test statistics did better than the modified versions. When there were 4 and 5 populations, the results varied.

**Conclusion:** All of the test statistics reached their asymptotic distributions quickly. The standardize first versions of the test statistics were generally better than the standardized last version of the test statistics, which meant that it was better to place equal weights on the RCBD portion and the CRD portion.

Open Access Original Research Article

Sule Ibrahim, Bello Olalekan Akanji, Lawal Hammed Olanrewaju

We propose a new distribution called the extended generalized inverse exponential distribution with four positive parameters, which extends the generalized inverse exponential distribution. We derive some mathematical properties of the proposed model including explicit expressions for the quantile function, moments, generating function, survival, hazard rate, reversed hazard rate and odd functions. The method of maximum likelihood is used to estimate the parameters of the distribution. We illustrate its potentiality with applications to two real data sets which show that the extended generalized inverse exponential model provides a better fit than other models considered.

Open Access Original Research Article

A. A. Sanusi, S. I. S. Doguwa, I. Audu, Y. M. Baraya

In this paper, we developed a new class of continuous distributions called Burr X Exponential-G Family. Also, we obtained sub-models of this family of distributions such as Burr X Exponential-Rayleigh (BXE-R) and Burr X Exponential Lomax (BXE-Lx) distributions; by showing their respective densities functions. Some structural properties of the proposed family of distributions were derived such as moment, moment generating function, probability weighted moment, renyi entropy and order statistics. We estimate the parameters of the model by using Maximum Likelihood methods. Finally, the results obtained are validated using two real data sets. The results show that BXE-Lx distribution provides better fit in the data sets than some other well known distributions. However, this new family of distributions will serve as an additional generator for developing new sub models to modeling positive real data sets.

Open Access Review Article

W. H. Moolman

The maximum flow and minimum cost-maximum flow problems are both concerned with determining flows through a network between a source and a destination. Both these problems can be formulated as linear programming problems. When given information about a network (network flow diagram, capacities, costs), computing enables one to arrive at a solution to the problem. Once the solution becomes available, it has to be applied to a real world problem. The use of the following computer software in solving these problems will be discussed: R (several packages and functions), specially written Pascal programs and Excel SOLVER. The minimum cost-maximum flow solutions to the following problems will also be discussed: maximum flow, minimum cost-maximum flow, transportation problem, assignment problem, shortest path problem, caterer problem.

Open Access Review Article

W. H. Moolman

The out-of-kilter algorithm, which was published by D.R. Fulkerson [1], is an algorithm that computes the solution to the minimum-cost flow problem in a flow network. To begin, the algorithm starts with an initial flow along the arcs and a number for each of the nodes in the network. By making use of Complementary Slackness Optimality Conditions (CSOC) [2], the algorithm searches for out-of-kilter arcs (those that do not satisfy CSOC conditions). If none are found the algorithm is complete. For arcs that do not satisfy the CSOC theorem, the flow needs to be increased or decreased to bring them into kilter. The algorithm will look for a path that either increases or decreases the flow according to the need. This is done until all arcs are in-kilter, at which point the algorithm is complete. If no paths are found to improve the system then there is no feasible flow. The Out-of-Kilter algorithm is applied to find the optimal solution to any problem that involves network flows. This includes problems such as transportation, assignment and shortest path problems. Computer solutions using a Pascal program and Matlab are demonstrated.