When you run an experiment with blocks, you are making the assumption that the blocks represent a sample of a population of possible blocks. Then you can draw conclusions about your treatments as related to the population of possible blocks when you use block as a random effect. If you use block as a fixed effect than your decisions about the treatments can be applied only to the set of blocks in your study---not a very interesting place to make inferences. The main place you will see using block as a random effect is in the standard errors of the means. The standard error of the mean will include the block variance component when blocks are considered as random and will not include the block variance component when the blocks are considered as fixed. So you may say treatment has a mean of 50 and a confidence interval of 48-52 for the fixed effect model. But when the blocks are considered to be random the confidence interval may be 42-58. It is larger as you are wanting to make inferences to the population of blocks. The comparisons of treatments that occur within a block will have smaller variances than comparisons that do not occur within blocks. If you have such an unconnected design you will not be able to estimate some treatment comparisons when the blocks are considered as fixed effects while a mixed models software will be able to provide treatment estimates and comparisons when blocks are declared as random.

Thanks a lot for your answer. I understand that the main advantage of using a random effect is for the (incomplete) blocks.

But what do you think about the replications (3 levels)?

Suppose that I would analyse my design as a RCBD (without including the blocks in the model), then I could classicaly consider replications as a fixed effect.

But is there a reason why replications should become random when using a mixed model that takes into account the alpha-lattice design, i.e. including a random block effect ?

My response has nothing to do with complete blocks or incomplete blocks. I am assuming that the term replication is used to denote a block with a set of main treatments and the incomplete blocks contain the secondary set of treatments. In any case replication or block or incomplete block are considered as a random effect so you can make inferences to a population of blocks the blocks in the study represent. If you run an experiment with complete blocks where each block is a replication of the treatments and you consider the levels of replication to be fixed then you can only make inferences about your treatments to the replications you have in your study. This is called a narrow inference space. If blocks are assumed to be random then you are making inferences about the treatments on a broad inference space. So we need to make a distinction between a block and a replicate. A block is a set of similar experimental units to which I randomly assign treatments. A replication is the independent observation of a treatment on different experimental units. Three experimental units per treatment provides three replications. Now suppose we have a completely randomized design where each treatment is assigned to three experimental units. There are three replications but only one block or no blocks. Next suppose we have three blocks each with the same number of experimental units which is the number of treatments. We would randomly assign each treatment to one experimental unit within each block. In this case the treatments within each block provide a complete replication of all of the treatments. This is where the confusion occurs in that a lot of authors call the block a replication, but it is a block that just happens to contain a complete replication of the treatments. As a reference, see Milliken and Johnson, Analysis of Messy Data Vol 1 2nd Edition, CRC Press, for more detailed explanation.

I wouldn't consider replicates at all in the model, just treatments and incomplete blocks within replicates (as random). If you want to know, out of curiosity, whether the whole replicates have an effect, you can run simple RCB analysis (with replicates as fixed, of course), but the goal of the experiment is to compare the treatments, with best possible precision, and that's what alpha-lattice design is supposed to give. Even if you wanted to include replicates in your mixed models as a random factor, with only three replicates I think you don't have enough degrees of freedom to get reliable estimate of the corresponding variance component.

BTW, why not consider (in the future) using row-columns designs (two-dimensional variant of alpha-lattice designs), which don't use more plots but provide control in two dimensions?

The rows of the alpha design are blocks and those blocks that should be the random effects---it does not make any difference as to the number of replications of each treatment. If you use row and column designs then the appropriate analysis involves declaring the rows and columns as random effects.

Absolutely, an appropriate statistical analysis model is required for each experimental design, and in this case rows and columns shoud be treated as random factors.

Hi , I have used simple alpha lattice design which includes 

136 BC1F2 families and 4 checks 

Replications: 3

10 blocks in each replication

14 BC1F2 families in each block.

blocks are taken columns , therefore 10 columns

families are taken as rows, therefore 42 rows.

For analysis which model i have to go for in SAAS since there are GLM procedure, MIxed models available,