Some New Dimensions to Construct Economical Circular Weakly Balanced Neighbor Robust Designs

Neighbor-balanced designs are used in the experiments where the performance of a treatment is aected by the treatments applied to its neighboring units. ese are well-known designs to balance the neighbor eects. Among these designs, minimal neighbor-balanced designs are economical; therefore, these are preferred by the experimenters. For v even, minimal neighbor balanced designs in circular blocks cannot be constructed formost of the cases, where v is number of treatments. In such situations, experimenters would like to relax the neighbor balance property up to some extent and consider the minimal circular weakly balanced neighbor designs as the better alternates to the minimal circular neighbor-balanced designs. In this article, some generators are developed to obtain minimal circular weakly balanced neighbor designs in blocks of equal, two and three dierent sizes.


Introduction
Neighbor-balanced designs (NBDs) are used in the experiments where performance of a treatment is in uenced by the treatments applied to its neighboring units. NBDs are well-known designs to balance the neighbor e ects. Among the NBDs, minimal NBDs are economical; therefore, these are preferred by the experimenters. If each pair of distinct treatments appears once in adjacent plots, then, the design is called minimal NBD. A block formed in a cycle in such a way that its rst and last units are considered as adjacent neighbors is called a circular block. In circular blocks, each unit has one left-neighbor and one right-neighbor. Williams [1] suggested nearest neighbor-balanced designs (NBDs) in linear blocks for rst-order autoregressive model. Rees [2] introduced neighbor designs in a technique used in virus research and constructed these designs for odd number of treatments (v). Lawless [3] gave a note on di erent types of balanced incomplete designs balanced for residual e ects. Hwang and Lin [4,5] constructed four classes of NBDs for v [6] discussed that the bias due to neighbor e ects can be minimized through NBDs. Azais et al. [6] suggested that NBDs should be used to deal with neighbor e ects and if a NBD requires large number of experimental units, a partially NBD is used. Jacroux [7] constructed equineighbored designs in blocks of sizes three. Kunert [8] showed that the bias due to neighbor e ects can be minimized through NBDs or partially NBDs. Ahmed and Akhtar [9] constructed some series of NBDs which are not minimal. Akhtar and Ahmed [10] constructed some second-order NBDs for some cases of v [11] constructed some all-order NBDs forV odd. Ahmed and Akhtar [12] constructed some NBDs in circular blocks of sizes six which are not minimal. Akhtar et al. [13] constructed some NBDs in circular blocks of sizes ve. Misra and Nutan [14,15] constructed circular generalized neighbor designs (CGNDs) by relaxing the neighbor balance property.
e CGNDs will become more economical and better alternate to the minimal CNBDs if most of the unordered pairs appear (i) once as neighbor while other appear no time, or (ii) once as neighbor while other appear twice. GNDs possessing this property are also called minimal circular weakly balanced neighbor designs (CWBNDs). Mishra [16] presented some families of proper GNDs. Ahmed et al. [17] constructed economical GNDs in use of serology. Zafaryab et al. and Shehzad et al. [18,19] developed some generators to obtain the minimal CWBNDs (MCWBNDs) for some cases. Iqbal et al. [20] presented GNDs for k (block size) � 3. is review shows that MCWBNDs-II (MCWBNDs in which 3 v/2 unordered pairs do not appear as neighbors while all other appear once) are not available in the literature. In this study, therefore, some generators are presented to generate MCWBNDs-II for all k > 2. ese designs are economical and efficient to balance the neighbor effects.
Neutrosophic statistics is the extension of classical statistics and is applied when the data are coming from a complex process or from an uncertain environment. e proposed study can be extended for neutrosophic statistics. Few researchers [21][22][23] constructed designs of (i) sampling plan for exponential distribution under the neutrosophic statistical interval method, (ii) a new attribute control chart under neutrosophic statistics, and (iii) variable sampling plan for Pareto distribution using the neutrosophic statistical interval method.

Method of Cyclic Shifts
is method was introduced by Iqbal [24] which is simplified here for CBNDs and MCWBNDs-II.
(i) If all from 1 to v-1 appears exactly once in S * , then designs will be MCBND.
(ii) If all from 1 to v-1 appears once in S * except any three elements which do not appear, then designs will be MCWBND-II. For S 1 , get v Blocks and assign 0, 1, . . ., v-1 to first units of each block, respectively. Add 4 (mod 18) to the first unit element of each block for second unit elements. Similarly, add 6 (mod 18) and so on (see Table 1).
Take v more blocks for S 2 and obtain the design (see Table 2). Table 1 and Table 2 jointly present the MCWBND-II for v � 18, k 1 � 4, and k 2 � 3. In this design, 27 unordered pairs do not appear as neighbors while all other appear once.

Logic behind Rule I.
Using the logic behind Rule I, following theorems are developed for MCWBNDs-II.  e efficiency factor for neighbor effect (residual effect) is the harmonic mean of nonzero Eigen values of their respective information matrix (see [25]). e high value of E n shows that design is suitable for the estimation of neighbor effects. Our developed generators provide the designs with high values of E n for v even; therefore, these designs are suitable for the estimation of neighbor effects. Table 3 shows the the E n of some existing MCBNDs and proposed designs for comparison.

Construction of MCWBNDs-II
Here, minimal CWBNDs-II are generated through Rule I. To generate required MCWBNDs-I, sets of shifts will be obtained from the elements of selected S, see eorem 2-eorem 4. Here, m � (v-2)/2, and i, u, and w are integers.

MCWBNDs-II in Blocks of Equal Sizes.
Here, MCWBNDs-II are generated in equal block sizes from i sets which will be generated in the following manner. Divide the elements of selected S in i groups of size k in such a way that v is the factor of sum of each group; then, discard one element from each group.

MCWBNDs-II in Blocks of Two Different Sizes.
Here, MCWBNDs-II are constructed in two different block sizes from (i+1) sets which will be generated in the following manner.
Divide the elements of selected S in i groups of size k 1 and one of k 2 in such a way that v is the factor of sum of each group; then, discard one element from each group.

MCWBNDs-II in Blocks of ree Different Sizes.
Here, MCWBNDs-II are constructed in three different block sizes using (i+2) sets which will be generated in the following manner.
Divide the elements of selected S in i groups of size k 1 and one each of k 2 and k 3 in such a way that v is factor of sum of each group; then, discard one element from each group.

Conclusions and Future Research
Neighbor effects may arise in experiments of serology for virus research and in agriculture experiments, due to nature of plots, etc. In the presence of neighbor effects, misleading conclusions may be drawn in the variety competition experiments. Minimal NBDs are available in the literature to neutralize these effects economically for v odd. To overcome this problem for v even, complete solution is given in this article to construct proposed MCWBNDs-I for each and every k > 2. At least v(v-1) experimental units are required to satisfy the neighbor balance property for v even. MCWBND-II requires vv(v-4)/2 units. Hence, we lose [3/(v − 1) × 100]% neighbor balance and save at least [50(v + 2)/v − 1]% experimental units. Our proposed designs save more than 50% of experimental material for v even at the cost of losing at most 3% efficiency of neighbour effects, therefore, are efficient to reduce the bias due to neighbor effects. As a future research, (i) an algorithm coded with R-language can be developed to generate the proposed designs, (ii) the proposed new designs can be applied in experiments of serology and agriculture to obtain the real data for comparison, and (iii) the current study can be extended using neutrosophic statistics [26].
Data Availability e data used in the article are included within the article.

Conflicts of Interest
e authors declare no conflicts of interest.